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# The chromatic number of the product of two 4-chromatic graphs is 4

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## Abstract

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs.

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... The complete graphs K 1 , K 2 , K 3 are known to be multiplicative (see [4]). Only the cases of K 4 , . . . ...
... admits a proper (n + 5)-colouring c defined by c ((u, v), f ) = f (u, v) (see [4]). Therefore to complete our example it only remains to prove the following. ...
... Thus, χ(M m (G)) = χ(G) if and only if K G n contains such a path of length m from a constant map to a proper n-colouring. In particular, χ(M m (K n )) = n + 1 for m ≥ 1 and n ≥ 2, since the proper n-colourings of K n are isolated vertices in K Kn n (see [4]). We next present a 4-chromatic graph T such that χ(M 3 (T )) = 4. ...
... So, the common wisdom suggests that for product-graphs with chromatic numbers three or more, Hedetniemi's conjecture should be refuted or proved in one fell swoop. In 1986, twenty years later after Hedetniemi's statement, El-Zahar and Sauer [ES85] proved that if the product of two graphs is 3-colourable, then one of the factors is 3-colourable. Nevertheless, the general case did not follow, and even the case of 4-colourings remains wide open. ...
... It is easy to show that the strongly multiplicative graphs are multiplicative, despite the extra assumption of connectedness. The main result of El-Zahar and Sauer [ES85] was that K 3 is multiplicative. In the concluding comments, they noted that their proof actually shows that K 3 is strongly multiplicative. ...
... There was no mention of strong multiplicativity in that paper, but nonetheless, the proof again establishes the stronger property. Therefore the following result can be be credited to [ES85] and [Häg+88]: ...
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A graph K is multiplicative if a homomorphism from any product G x H to K implies a homomorphism from G or from H. Hedetniemi's conjecture states that all cliques are multiplicative. In an attempt to explore the boundaries of current methods, we investigate strongly multiplicative graphs, which we define as K such that for any connected graphs G,H with odd cycles C,C', a homomorphism from $(G \times C') \cup (C \times H) \subseteq G \times H$ to K implies a homomorphism from G or H. Strong multiplicativity of K also implies the following property, which may be of independent interest: if G is non-bipartite, H is a connected graph with a vertex h, and there is a homomorphism $\phi \colon G \times H \to K$ such that $\phi(-,h)$ is constant, then H admits a homomorphism to K. All graphs currently known to be multiplicative are strongly multiplicative. We revisit the proofs in a different view based on covering graphs and replace fragments with more combinatorial arguments. This allows us to find new (strongly) multiplicative graphs: all graphs in which every edge is in at most square, and the third power of any graph of girth >12. Though more graphs are amenable to our methods, they still make no progress for the case of cliques. Instead we hope to understand their limits, perhaps hinting at ways to further extend them.
... Theorem 3.1 Let k ≥ 5 be an integer. Let G and H be graphs with min{χ(G), χ(H)} = k, and suppose that (8,9), (8,10), (9,8), (10,8)}. ...
... Theorem 3.1 Let k ≥ 5 be an integer. Let G and H be graphs with min{χ(G), χ(H)} = k, and suppose that (8,9), (8,10), (9,8), (10,8)}. ...
... Theorem 3.1 Let k ≥ 5 be an integer. Let G and H be graphs with min{χ(G), χ(H)} = k, and suppose that (8,9), (8,10), (9,8), (10,8)}. ...
Preprint
For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. In this paper, we give a reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals of polynomial rings, and we demonstrate computational experiments for partial solutions of Hedetniemi's conjecture along such a strategy using Gr\"{o}bner basis.
... That K 2 is multiplicative -i.e., a product of two graphs is bipartite iff one of the factors isfollows easily from the fact that a graph is bipartite iff it has no odd-length cycle. K 3 was proved to be multiplicative by El Zahar and Sauer [ES85]. Their proof was generalized to odd cycles by Häggkvist et al. [Häg+88]. ...
... Then [C] → [P ] · [C] · [P ] −1 is easily checked to be a group isomorphism between π v (K p/q ) /∼ and π 0 (K p/q ) /∼ . Next, with give a very short proof of a parity argument used in [ES85;Häg+88;DS02]. For a half-parity cycle C in G or H, if [µ(C ⊗ h 0 h 1 )] = X ·2 for some X ∈ π(K) /∼ , define the half-parity of C as the parity of |X|. ...
... Finally, we use what we obtained to get a graph homomorphism G → K, similarly as in [ES85], except for using the relaxed condition on edges instead of a condition on vertices. ...
Article
A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In particular, this gives the first multiplicative graphs of chromatic number higher than 4. Generalizing, in terms of the box complex, the topological insight behind existing proofs for odd cycles, we also give a different proof for circular cliques.
... This proves some cases of Hedetniemi's conjecture of 1966 [6], which states that (2) χ(G × H) = min{χ(G), χ(H)}. ...
... The graphs and chromatic numbers involved in the counterexamples to Hedetniemi's conjecture discovered by Shitov were extremely large. Then in 2020, Zhu [16] found smaller counterexamples to (2), where χ(G × H) = 125. These are also counterexamples to (3). ...
... In this paper, we adapt the examples of [16] to show that inequality (3) can fail with χ(G × H) = 49. The strongest positive results on Hedetniemi's conjecture are those of El-Zahar and Sauer [2], who proved that (2) holds whenever χ(G × H) ≤ 3. So, all that remains to find out is whether the categorical product of 5-chromatic graphs can be 4-chromatic. This last remaining open case of Hedetniemi's conjecture can be weakened considerably, to a fractional form in the spirit of (3): It would be interesting to know if and how the condition χ f (G) > B can be used as a "strong structural property" yielding a similar result-though considerably weaker-at least for the case χ(G × H) > 4. ...
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We show that the inequality χ(G × H) < min{χ f (G), χ(H)} can happen when χ(G × H) = 49, improving on the lowest previously known value χ(G × H) = 125.
... A standard tool in the study of Hedetniemi's conjecture is the concept of the exponential graph as introduced in [3]. Let c be a positive integer, and let H be a finite graph that we allow to contain loops; the graph E c (H) has all mappings V (H) → {1, . . . ...
... The relevance of E c (H) to the problem comes from the fact that the graph H × E c (H) has a proper c-coloring defined as (h, ψ) → ψ(h). Another basic result in [3] tells that the constant mappings form a c-clique in E c (H), which means that these mappings get different colors in a c-coloring. Without loss of generality, we assume that the color labels of a proper coloring Ψ : E c (H) → {1, . . . ...
... We set c = ⌈4.1q⌉ and pass to sufficiently large q; we immediately get χ(G⊠K q ) q ·χ f (G) > c and also χ (E c (G ⊠ K q )) > c by Claim 4. The equality χ ((G ⊠ K q ) × E c (G ⊠ K q )) = c follows by standard theory [3]. ...
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The chromatic number of $G\times H$ can be smaller than the minimum of the chromatic numbers of finite simple graphs $G$ and $H$.
... Hedetniemi's conjecture and multiplicative graphs Hedetniemi [Hed66] more than 50 year ago conjectured the following: χ(G×H) = min(χ(G), χ(H)), for any graphs G, H. (Here × is the tensor, or categorical product and χ is the chromatic number, see Section 2 for definitions). Despite the simplicity of the statement, very little is known: one of the strongest results is a proof by El-Zahar and Sauer [ES85] that the conjecture is true when ...
... graph when G × H → K implies G → K or H → K, for all graphs G, H. Hedetniemi's conjecture is then that all clique graphs K n are multiplicative. El-Zahar and Sauer's [ES85] result amounts to saying that K 3 is multiplicative. This has been generalized to odd cycles by Häggkvist et al. [Häg+88], to circular cliques K p/q with p/q < 4 by Tardif [Tar05] (using iterations of the Γ 3 and Ω 3 functors) and to all graphs with no 4-cycles by the author [Wro17] (using the box complex). ...
... The proof of the multiplicativity of K 3 by El-Zahar and Sauer [ES85], its generalization to odd cycles by Häggkvist et al. [Häg+88], and especially its reformulation and generalization to circular cliques K p/q (with 2 < p/q < 4) given in [Wro17], largely follows the steps of the above proof of Lemma 3.3. An invariant on odd cycles is considered, which turns out to be exactly the winding number assigned as above to the corresponding map S 1 → Z 2 |Box(G)|. ...
Article
We consider a natural graph operation Ωk that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the Z2-homotopy type) of the box complex, a basic tool in applications of topology in combinatorics. Moreover, we prove that the box complex of a graph G admits a Z2-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph Ωk(G) admits a homomorphism to H, for high enough k. This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products is true, then the following analogous conjecture in topology is also true: If n∈N and X,Y are Z2-spaces (finite Z2-simplicial complexes) such that X×Y admits a Z2-map to the n-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture.
... In 1966, Hedetniemi conjectured in [7] that equality always hold in the above inequality. This conjecture received a lot of attention in the past half century (see [1,8,11,13,18,19]). Some special cases are confirmed. ...
... Some special cases are confirmed. In particular, it was proved by El-Zahar and Sauer [1] that Hedetniemi's conjecture holds for c = 3 (where for c ≤ 2, the conjecture holds trivially). Also, it was proved in [19] that a fractional version of Hedetniemi's conjecture is true, i.e., for any graphs G and H, χ f (G × H) = min{χ f (G), χ f (H)}. ...
... Note that a homomorphism from a graph G to K c is equivalent to a proper c-colouring of G. Thus if G → H, then χ(G) ≤ χ(H). The following result was proved in [1]. For the completeness of this paper, we include a short proof. ...
Preprint
Hedetniemi conjectured in 1966 that $\chi(G \times H) = \min\{\chi(G), \chi(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $(3+4/(p-1))$. Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number about $p^22^{p+1}$ and with more than $(p^22^{p+1})^{p^32^{p-1}}$ vertices. In this paper, we show that the conjecture fails already for some graphs $G$ and $H$ with chromatic number $3(p-1)/2 +3$ and with at most $\left(3 \lceil \frac {p-1}2 \rceil +2\right)p+ 3 \lceil \frac{p-1}2 \rceil + 3$ vertices. The currently known upper bound for $p$ is $607$ (and we expect the exact value of $p$ to be much smaller). Thus Hedetniemi's conjecture fails for some graphs $G$ and $H$ with chromatic number $912$, with at most $553,889$ vertices.
... This conjecture received a lot of attention in the past half century (see [1,9,12,16,21,22]). Some special cases are confirmed. ...
... Some special cases are confirmed. In particular, it was proved by El-Zahar and Sauer [1] that Hedetniemi's conjecture holds for c = 3 (where for c ≤ 2, the conjecture holds trivially). Also, it was proved in [22] that a fractional version of Hedetniemi's conjecture is true, i.e., for any graphs G and H, ...
... Note that a homomorphism from a graph G to K c is equivalent to a proper c-colouring of G. Thus if G → H, then χ(G) ≤ χ(H). The following result was proved in [1]. For the completeness of this paper, we include a short proof. ...
Article
Hedetniemi conjectured in 1966 that χ(G×H)=min⁡{χ(G),χ(H)} for all graphs G and H. Here G×H is the graph with vertex set V(G)×V(H) defined by putting (x,y) and (x′,y′) adjacent if and only if xx′∈E(G) and yy′∈E(H). This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let p be the minimum number of vertices in a graph of odd girth 7 and fractional chromatic number greater than 3+4/(p−1). Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number about p33p. In this paper, we show that the conjecture fails already for some graphs G and H with chromatic number 3⌈p+12⌉ and with p⌈(p−1)/2⌉ and 3⌈p+12⌉(p+1)−p vertices, respectively. The currently known upper bound for p is 83. Thus Hedetniemi's conjecture fails for some graphs G and H with chromatic number 126, and with 3,403 and 10,501 vertices, respectively.
... The conjecture that the first statement is true was formulated by Hedetniemi [Hed66]. Exponential graphs were used by El-Zahar and Sauer to prove the first nontrivial case of Hedetniemi's conjecture: the categorical product of two 4-chromatic graphs is 4-chromatic [ES85]. Tardif later converted this proof into an algorithm for the following problem: Given a graph H such that χ(H) > 3, find a proper 3-coloring of K H 3 [Tar06]. ...
... Based on the results of El-Zahar and Sauer [ES85], Tardif showed how to properly 3-color the graph K C 2n+1 even [Tar06]. Let ab denote a fixed (but arbitrarily chosen) edge in C 2n+1 . ...
... We note that Tardif uses the main result from [ES85] to show that B 2n+1 is bipartite. Our algorithm gives another proof that B 2n+1 is bipartite, and consequently, we give an alternative proof of the main result of [ES85]. ...
Preprint
Let $H=(V,E)$ denote a simple, undirected graph. The 3-coloring exponential graph on $H$ is the graph whose vertex set corresponds to all (not necessarily proper) 3-colorings of $H$. We denote this graph by $K_3^H$. Two vertices of $K_3^H$, corresponding to colorings $f$ and $g$ of $H$, are connected by an edge in $K_3^H$ if $f(i) \neq g(j)$ for all $ij \in E$. El-Zahar and Sauer showed that when $H$ is 4-chromatic, $K_3^H$ is 3-chromatic~\cite{el1985chromatic}. Based on this work, Tardif gave an algorithm to (properly) 3-color $K_3^H$ whose complexity is polynomial in the size of $K_3^H$~\cite{tardifAlg}. Tardif then asked if there is an algorithm in which the complexity of assigning a color to a vertex of $K_3^H$ is polynomial in the size of $H$. We present such an algorithm, answering Tardif's question affirmatively.
... A famous conjecture about the chromatic number of the direct product of graphs is the Hedetniemi's Conjecture [11], which stated that the above bound is tight, i.e., that χ(G × H) = min{χ(G), χ(H)}. This holds for graphs with chromatic number at most 4 [7], but the conjecture remained open for 52 years until it was recently proved false by Shitov [22]. Here, we present a result similar to the one in [7], proving that equality holds for digraphs with dichromatic number at most 2. ...
... This holds for graphs with chromatic number at most 4 [7], but the conjecture remained open for 52 years until it was recently proved false by Shitov [22]. Here, we present a result similar to the one in [7], proving that equality holds for digraphs with dichromatic number at most 2. ...
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The dichromatic number of a digraph $G$ is the smallest integer $\chi_a(G)$ such that the vertex set of $G$ can be partitioned into $\chi_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of digraphs: $\chi_a(G\square H)=\max\{\chi_a(G),\chi_a(H)\}$; $\chi_a(G\times H)\le \min\{\chi_a(G),\chi_a(H)\}$; and $\chi_a(G[H]) = \chi_a(G[\overset{\leftrightarrow}{K}_k])$, where $k =\chi_a(H)$ and $\overset{\leftrightarrow}{K}_k$ denotes the complete digraph on $k$ vertices. In addition, we investigate the products of directed cycles, giving exact values for $\chi_a(\overset{\rightarrow}{C}_n\times \overset{\rightarrow}{C}_m)$ and $\chi_a(\overset{\rightarrow}{C}_n\boxtimes \overset{\rightarrow}{C}_m)$ for every $n,m$, and for $\chi_a(\overset{\rightarrow}{C}_n[H])$ for every positive integer $n$. This latter result generalizes a result given in \cite{PP.16}, where they give exact values when $n>\chi_a(H)$. We also provide a upper-bound to the dichromatic number of a digraph $G$ as a function of the treewidth of its underlying graph and we present an {\FPT}-time algorithm that computes the dichromatic number of $G$, when parameterized by treewidth of the underlying graph of $G$.
... It is easy to prove that Conjecture 1.1 is true for graphs of chromatic numbers 2 and 3. In [12], El-Zahar and Sauer, showed that the chromatic number of the product of two 4-chromatic graph is 4. The following are the some of the results in the support of the Conjecture 1.1. ...
... Proof of Theorem 1.11. In [12], El-Zahar and Sauer (see also Proposition 2.2, [30] showed that the Conjecture 1.1 is equivalent to the statement that χ(K A m ) = m for all m < χ(A). They first proved and then used the fact that for any graph homomorphism φ : A×B → K m , there are induced graph homomorphisms φ A : A → K B m defined by φ A (a)(b) = φ(a, b) and φ(a, b). ...
Preprint
The neighborhood complex $\N(G)$ of a graph $G$ were introduced by L. Lov{\'a}sz in his proof of Kneser conjecture. He proved that for any graph $G$, \begin{align} \label{abstract} \chi(G) \geq conn(\N(G))+3. \end{align} In this article we show that for a class of exponential graphs the bound given in (\ref{abstract}) is sharp. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs. Hedetniemi's conjecture states that the chromatic number of the categorical product of two graphs is the minimum of the chromatic number of the factors. Let $M(G)$ denotes the Mycielskian of a graph $G$. We show that, for any graph $G$ containing $M(M(K_n))$ as a subgraph and for any graph $H$, if $\chi(G \times H) = n+1$, then $\min\{\chi(G), \chi(H)\} = n+1$. Therefore, we enrich the family of graphs satisfying the Hedetniemi's conjecture.
... It is easy to prove that Conjecture 1.7 is true for graphs of chromatic numbers 2 and 3. In [11], El-Zahar and Sauer, showed that the chromatic number of the product of two 4-chromatic graph is 4. The following results give us some more classes of graphs for which Conjecture 1.7 is true. ...
... 3.3 Proof of the result of Sect. 1.3Proof of Theorem 1.11 In[11], El-Zahar and Sauer (see also Proposition 2.2,[28]) showed that the Conjecture 1.7 is equivalent to the statement that vðK A m Þ ¼ m for all m\vðAÞ. They first proved and then used the fact that for any graph homomorphism / : A Â B ! K m , there are induced graph homomorphisms / A : A ! K B m defined by / A ðaÞðbÞ ¼ /ða; bÞ and / B : B ! K A m defined by / B ðbÞðaÞ ¼ /ða; bÞ. ...
Article
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The neighborhood complex N(G) of a graph G was introduced by L. Lovász in his proof of Kneser conjecture. He proved that for any graph G, 2χ(G)≥conn(N(G))+3.In this article we show that for a class of exponential graphs the bound given in (2) is tight. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs. In 1966, Hedetniemi conjectured that the chromatic number of the categori-cal product of two graphs is the minimum of the chromatic number of the factors. In 2019, Shitov [26] gave a counterexample to this conjecture. Let M(G) denotes the Mycielskian of a graph G. We show that, for any graph G containing M(M(Kn)) as a subgraph and for any graph H, if χ(G×H)=n+1, then min{χ(G),χ(H)}=n+1. Therefore, we enrich the family of graphs satisfying the Hedetniemi’s conjecture.
... As already proved by El-Zahar & Sauer [ES85], one can assume without loss of generality that H is the exponential graph K G c , or a subgraph thereof, if one seeks to minimise |V (H)| or |E(H)|. Zhu [Zhu20] simplified Shitov's proof, giving an explicit construction of H (pointing to explicit vertices in the exponential graph, instead of relying on an asymptotic argument on their existence). ...
... In fact K G c is the most general such graph: for any graph H, we have G × H → K c if and only if H → K G c . Thus to give a counterexample for c-colorings, it suffices [ES85] to look for G such that χ(G) > c and χ(K G c ) > c. ...
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Hedetniemi's conjecture~\cite{hedetniemi1966homomorphisms} for $c$-colorings states that the tensor product $G \times H$ is $c$-colorable if and only if $G$ or $H$ is $c$-colorable. El-Zahar and Sauer~\cite{El-ZaharS85} proved it for $c = 3$. In a recent breakthrough, Shitov~\cite{Shitov19} showed counterexamples, for large $c$. While Shitov's proof is already remarkably short, Zhu \cite{Zhu20} simplified the argument and gave a more explicit counterexample for $c=125$. Tardif \cite{Tardif20} showed that a modification of the arguments allows to use wide colorings'' to obtain counterexamples for $c=14$, and $c=13$ with a more involved use of lexicographic products. This note presents two more small modifications, resulting in counterexamples for $c=5$ (with $G$ and $H$ having 4686 and 30 vertices, respectively).
... It is clear that H(1) and H(2) are true. El-Zahar and Sauer [6] show that H(3) is true. Thus Theorem 1.2 gives the following corollary. ...
... As is stated in Section 1, Corollary 1.4 follows from Theorem 1.2 and [6], The purpose of this section is to give a direct proof of Corollary 1.4. In fact, it is not necessary to assume that Z 2 -complexes X and Y in Corollary 1.4 are finite. ...
Article
The $\mathbb{Z}_2$-index ${\rm ind}(X)$ of a $\mathbb{Z}_2$-CW-complex $X$ is the smallest number $n$ such that there is a $\mathbb{Z}_2$-map from $X$ to $S^n$. Here we consider $S^n$ as a $\mathbb{Z}_2$-space by the antipodal map. Hedetniemi's conjecture is a long standing conjecture in graph theory concerning the graph coloring problem of tensor products of finite graphs. We show that if Hedetniemi's conjecture is true, then ${\rm ind}(X \times Y) = \min \{ {\rm ind}(X) , {\rm ind}(Y)\}$ for every pair $X$ and $Y$ of finite $\mathbb{Z}_2$-complexes.
... What if G and G ′ are subgraphs of G S ? A fundamental tool for the investigation of Hedetniemi's conjecture is the n-coloring graph C n (G) of a graph G defined by El-Zahar and Sauer [5]. It is not clear, however, whether there exist analogous well-behaving objects for Borel graphs. ...
... Let G be a Borel graph. Define a graph C n (G) of n-colorings of G for which the results of El-Zahar and Sauer [5] can be generalized. ...
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We show that there is no simple (e.g. finite or countable) basis for the Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift-graph on $[\mathbb{N}]^\mathbb{N}$ with finite Borel chromatic number form a $\mathbf{\Sigma}^1_2$-complete set. This answers a question of Kechris and Marks and strengthens several earlier results. We also formulate a general theorem that can be used to show that certain ideals of Borel sets are $\mathbf{\Sigma}^1_2$-hard and give a counterexample to the (lightface) $\Delta^1_1$ analogue of Hedetniemi's conjecture.
... The complete graphs K 1 and K 2 can be shown to be multiplicative both in G and D with relatively straightforward arguments. The first nontrivial case of Hedetniemi's conjecture was established by El-Zahar and Sauer in the aptly named paper [5]. ...
... Then, recursively, the graphs n T (3) (K 3 ) are all multiplicative in G. It turns out that The proof given in [11] is an adaptation of the proof of Theorem 3.1 given in [5]. However it is possible to derive the multiplicativity of all odd cycles from that of K 3 using the following adjoint functors that generalize T (3) and T (3) : Let T (2k + 1) = (K 1 , P 2k+1 , 1 , 2 ), where P 2k+1 is the path with 2k + 1 edges and 1 , 2 map K 1 to the endpoints of P 2k+1 . ...
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We survey results on Hedetniemi's conjecture which are connected to adjoint functors in the "thin" category of graphs, and expose the obstacles to extending these results.
... In 1966, Hedetniemi [8] conjectured that equality always holds in (1). This conjecture has received a considerable amount of attention; for instance, it was proved if G and H are 4-colorable [3], if every vertex in G is contained in a large clique [1], or if G and H are Kneser graphs or hypergraphs [6]. Additionally, many natural variants of this conjecture have been studied. ...
... A pair of the form (G, E c (G)) is a natural candidate for counterexamples to Hedetniemi's conjecture. Indeed, El-Zahar and Sauer [3] observed that if χ(G × H) < min{χ(G), χ(H)} for some H, then this also holds for H = E c (G) where c = χ(G) − 1. In any case, it is easy to establish an upper bound on χ(G × H) when H = E c (G). ...
Preprint
Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant $\alpha>0$ such that for all $c$ sufficiently large, there exist graphs $G$ and $H$ with chromatic number at least $(1+\alpha)c$ for which $\chi(G \times H) \le c$.
... However, the difficulties of the set-system approach to hypergraphs are resolved by R, and many graph theoretic results have already been generalized to hypergraphs via "oriented hypergraphs" in [6], [7], [23], [24], [25], [26], [29]. The nature of quiver and graph exponentials is of particular importance regarding Hedetniemi's conjecture, where in [11] it was shown that the multiplicativity of K is equivalent to either G or K G is K-colorable, for every graph G. The main results provide structure theorems which illustrate; (1) that the difficulty in the set-system hypergraphic approach to generalizations of graph theory are categorical in nature; (2) the idiosyncrasies of set-systems are remedied in the category of incidence structures; (3) incidence structures are a faithful generalization of quivers via a logical functor; and (4) a characterization of the edges of quiver exponentials as morphisms under the logical inclusion into the category of incidence structures. ...
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This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e. quivers). The category of incidence hypergraphs is introduced as a "graph-like" remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is shown that the category of quivers embeds into the category of incidence hypergraphs via a logical functor that is the inverse image of an essential geometric morphism between the topoi. Consequently, the quiver exponential is shown to be simply represented using incidence hypergraph homomorphisms.
... In 1966, Hedetniemi conjectured that equality holds, and only a few special cases have been proven in the time since then. Most notably, El-Zahar and Sauer [3] gave a proof for when the minimum is four (for smaller values of the minimum, the proof is straightforward), but for all larger values the conjecture remains open. More recently, it was shown by Zhu [13] that the conjecture holds when chromatic number is replaced by fractional chromatic number. ...
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A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of $G$ is the smallest real number $t \ge 1$ for which a vector $t$-coloring of $G$ exists. For a graph $H$ and a vector $t$-coloring $p_1,\ldots,p_n$ of a graph $G$, the assignment $(i,\ell) \mapsto p_i$ is a vector $t$-coloring of the categorical product $G \times H$. It follows that the vector chromatic number of $G \times H$ is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lov\'{a}sz theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.
... The product of graphs G on X and G ′ on X ′ is the graph on X × X ′ given by ((x, x ′ ), (y, y ′ )) ∈ G × G ′ ⇐⇒ (x, y) ∈ G and (x ′ , y ′ ) ∈ G ′ . The Borel version of Hedetniemi's conjecture reads as follows: Is it the case that χ B (G × G ′ ) = min{χ B (G), χ B (G ′ )}? Theorem 1.1 implies that the answer is affirmative, if min{χ B (G), χ B (G ′ )} ≤ 3. El-Zahar and Sauer [9] showed that for finite graphs the bound 4 already implies an affirmative answer. Hence the following problem is quite natural. ...
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We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.
... The Hhomomorphism graph is perhaps better known as the reflexive subgraph of the exponential graph H G . Exponential graphs were introduced by Lovász in [22] and have been used to study and solve many interesting homomorphism and colouring problems, see for example [9,17]. For loop-free graphs, the connected components of C H (G) correspond to mixing classes, whereas the connected components of H H (G) correspond to homotopy classes. ...
Article
This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the -colorings of a graph G can be obtained by successively recoloring a single vertex provided along the lines of Cereceda, van den Heuvel, and Johnson's result for k-colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson-type extension theorem for -precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.
... Hedetniemi conjectured in 1966 that χ(G × H) = min{χ(G), χ(H)} for all finite graphs G and H [3]. The conjecture received a lot of attention [4,7,10,11] and remained open for more than half century. It is known that χ(G × H) = min{χ(G), χ(H)} whenever min{χ(G), χ(H)} ≤ 4 [1] and that the fractional version is true, i.e., for any graphs G and H, χ f (G × H) = min{χ f (G), χ f (H)} [12]. However, Shitov refuted this conjecture recently [9]. ...
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We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$.
... Hedetniemi's conjecture formulated in 1966 thus stated that K c is multiplicative for every positive integer c. This is trivial for c = 1, easy for c = 2 and is a far from trivial result by El-Zahar and Sauer [2] for c = 3 published in 1985. For no other c it was decided (whether K c is multiplicative or not) until 2019, when a breakthrough by Yaroslav Shitov took place who proved in [16] that the conjecture is not true by constructing counterexamples for large enough c's. ...
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A coloring is called $s$-wide if no walk of length $2s-1$ connects vertices of the same color. A graph is $s$-widely colorable with $t$ colors if and only if it admits a homomorphism into a universal graph $W(s,t)$. Tardif observed that the value of the $r^{\rm th}$ multichromatic number $\chi_r(W(s,t))$ of these graphs is at least $t+2(r-1)$ and equality holds for $r=s=2$. He asked whether there is equality also for $r=s=3$. We show that $\chi_s(W(s,t))=t+2(s-1)$ for all $s$ thereby answering Tardif's question. We observe that for large $r$ (with respect to $s$ and $t$ fixed) we cannot have equality and that for $s$ fixed and $t$ going to infinity the fractional chromatic number of $W(s,t)$ also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
... Hedetniemi conjectured in 1966 [3] that f (n) = n for all positive integer n. This conjecture received a lot of attention [1,4,8,11,13,14] and is disproved recently by Shitov in [9]. For a positive integer c and a graph G, let [c] = {1, 2, . . . ...
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The Poljak-R\"{o}dl function is defined as $f(n) = \min\{\chi(G \times H): \chi(G)=\chi(H)=n\}$. This note proves that $\lim_{n \to \infty} \frac{f(n)}{n} \le \frac 12$.
... The conjecture received a lot of attention [7,10,13,14] and remained open for more than half century. It is known that χ(G×H) = min{χ(G), χ(H)} whenever min{χ(G), χ(H)} 4 [1] and that the fractional version is true, i.e., for any graphs G and H, χ f (G × H) = min{χ f (G), χ f (H)} [15]. However, Shitov refuted this conjecture recently [12]. ...
Article
We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$.
... For n ≥ 1, define H n := G 1 × · · · × G n . By [EZS85] we get that the graphs H n are not 3colorable, which gives item (1). Item (2) follows since H n+1 projects into H n for all n ≥ 1. ...
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The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to the dichotomy conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure within the range of the conjecture whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of non-trivial height 1 identities differ for $\omega$-categorical structures with less than doubly exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.
... which is equivalent to every complete graph being multiplicative in the category of simple graphs. It was shown [15] that the multiplicativity of is equivalent to either or the exponential graph being -colorable, for every graph , reinforcing the necessity to study exponentials and strive for a natural approach to homs. ...
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The box product and its associated box exponential are characterized for the categories of quivers (directed graphs), multigraphs, set system hypergraphs, and incidence hypergraphs. It is shown that only the quiver case of the box exponential can be characterized via homs entirely within their own category. An asymmetry in the incidence hypergraphic box product is rectified via an incidence dual-closed generalization that effectively treats vertices and edges as real and imaginary parts of a complex number, respectively. This new hypergraphic box product is shown to have a natural interpretation as the canonical box product for graphs via the bipartite representation functor, and its associated box exponential is represented as homs entirely in the category of incidence hypergraphs; with incidences determined by incidence-prism mapping. The evaluation of the box exponential at paths is shown to correspond to the entries in half-powers of the oriented hypergraphic signless Laplacian matrix.
... admits a proper 14-colouring c defined by c((u, v), f ) = f (u, v) (see [2]). Therefore to complete our example it only remains to prove the following. ...
... The product of graphs on and ′ on ′ is the graph on × ′ given by (( , ′ ), ( , ′ )) ∈ × ′ ⇐⇒ ( , ) ∈ and ( ′ , ′ ) ∈ ′ . The Borel version of Hedetniemi's conjecture reads as follows: is ( × ′ ) = min{ ( ), ( ′ )}? Theorem 1.1 implies that the answer is affirmative when min{ ( ), ( ′ )} ≤ 3. El-Zahar and Sauer [9] showed that, for finite graphs, the bound 4 already implies an affirmative answer. Hence the following problem is quite natural. ...
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We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.
... Though not true in general, this holds in several special cases. In particular, it is easy to prove when min{χ(F ), χ(G)} ≤ 3 and it is also known to hold when this value is 4. The latter, however, is a highly nontrivial result of El Zahar and Sauer [11] and the general case was wide open until the already mentioned recent breakthrough by Shitov [30]. For several related results, see the survey papers [28,33,35]. ...
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Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter satisfying $C_{\rm OR}(F\times G)\le\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less, than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.
... In particular, the identity G × K G c → K c always holds. Hence Hedetniemi's conjecture is equivalent to the statement that if χ(G) > c, then χ(K G c ) ≤ c (see [7]). We denote K G the class of exponential graphs K G , G ∈ G, quotiented by ↔. ...
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We prove that for any c ≥ 5, there exists an infinite family (Gn) n∈N of graphs such that χ(Gn) > c for all n ∈ N and χ(Gm × Gn) ≤ c for all m = n. These counterexamples to Hedetniemi's conjecture show that the Boolean lattices of exponential graphs with Kc as a base are infinite for c ≥ 5.
Chapter
This paper presents brief discussions of ten of my favorite, well-known, and not so well-known conjectures and open problems in graph theory, including (1) the 1963 Vizing’s Conjecture about the domination number of the Cartesian product of two graphs [47], (2) the 1966 Hedetniemi Conjecture about the chromatic number of the categorical product of two graphs [28], (3) the 1976 Tree Packing Conjecture of Gyárfás and Lehel [23], (4) the 1981 Path Partition Conjecture of Lovász and Mihók [8], (5) the 1991 Inverse Domination Conjecture of Kulli and Sigarkanti [34], (6) the 1995 Queens Domination Conjecture [15], (7) the 1995 Nearly Perfect Bipartition Problem [9], (8) the 1998 Achromatic-Pseudoachromatic Tree Conjecture [10], (9) the 2004 Iterated Coloring Problems and the Four-Color Theorem [30], and (10) the 2011 γ-graph Sequence Problem [16].
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We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n. Our construction is an adaptation of the construction of counterexamples to Hedetniemi’s conjecture devised by Shitov, and adapted by Zhu to graphs with relatively small chromatic numbers. The new tools we introduce are graphs with minimal colourings that are “wide” in the sense of Simonyi and Tardos, and generalised Mycielskians to settle the case n = 13.
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A coloring is called s-wide if no walk of length 2 s − 1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W ( s , t ) . Tardif observed that the value of the r th multichromatic number χ r ( W ( s , t ) ) of these graphs is at least t + 2 ( r − 1 ) and equality holds for r = s = 2. He asked whether there is equality also for r = s = 3. We show that χ s ( W ( s , t ) ) = t + 2 ( s − 1 ) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W ( s , t ) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
Chapter
Although any isomorphism between two graphs is a homomorphism, the study of homomorphisms between graphs has quite a different flavour to the study of isomorphisms. In this chapter we support this claim by introducing a number of topics involving graph homomorphisms. We consider the relationship between homomorphisms and graph products, and in particular a famous unsolved conjecture of Hedetniemi, which asserts that if two graphs are not n-colourable, then neither is their product. Our second major topic is the exploration of the core of a graph, which is the minimal subgraph of a graph that is also a homomorphic image of the graph. Studying graphs that are equal to their core leads us to an interesting class of graphs first studied by Andrásfai. We finish the chapter with an exploration of the cores of vertex-transitive graphs.
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Hedetniemi conjectured in 1966 that if G and H are finite graphs with chromatic number n, then the chromatic number of the direct product of G and H is also n. We mention two well-known results pertaining to this conjecture and offer an improvement of the one, which partially proves the other. The first of these two results is due to Burr et al. (Ars Combin 1 (1976), 167–190), who showed that when every vertex of a graph G with is contained in an n-clique, then whenever . The second, by Duffus et al. (J Graph Theory 9 (1985), 487–495), and, obtained independently by Welzl (J Combin Theory Ser B 37 (1984), 235–244), states that the same is true when G and H are connected graphs each with clique number n. Our main result reads as follows: If G is a graph with and has the property that the subgraph of G induced by those vertices of G that are not contained in an n-clique is homomorphic to an -critical graph H, then . This result is an improvement of the result by the first authors. In addition we will show that our main result implies a special case of the result by the second set of authors. Our approach will employ a construction of a graph F, with chromatic number , that is homomorphic to G and H.
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We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be interpreted as a generalized uniform hypergraph where each edge has cardinality $\#X$ incident vertices (including multiplicity) and where the monoid informs what type of cohesivity the edges possess. One distinguishing feature of $(X,M)$-graphs is the presence of unfixed edges. We prove that unfixed edges are a necessary feature of a category of graphs or uniform hypergraphs if one wants exponentials and effective equivalence relations to exist in the category. The main advantage of separating syntax (the $(X,M)$-graph theories) from semantics (the categories of $(X,M)$-graphs) is the ability to interpret the theory in any cocomplete category. This interpetation functor then yields a nerve-realization adjunction and allows us to transfer structure between the category of $(X,M)$-graphs and the receptive cocomplete category.
Article
We prove that the coindex of the box complex of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to H. As a consequence, we exhibit for every graph H a system of linear equations, solvable in polynomial time, with the following properties: if the system has no solutions, then ; if the system has solutions, then . We generalise the method to other bounds on chromatic numbers using linear algebra.
Chapter
Direct ProductWreath ProductA Very Strong ProductGallai's Problem on Dirac's ConstructionHajós Versus OreLength of Hajós ProofsHajós Constructions of Critical GraphsConstruction of Hajós Generalized by DiracFour-Chromaticity in Terms of 3-Colorability
Chapter
List-Coloring Bipartite GraphsList-Coloring the Union of GraphsCochromatic NumberStar Chromatic NumberHarmonious Chromatic NumberAchromatic NumberSubchromatic NumberMultiplicative GraphsReducible Graph PropertiesT-ColoringsGame Chromatic NumberHarary and Tuza's Coloring GamesColoring Extension GameWinning Hex
Chapter
Hedetniemi conjectured in 1966 that Hedetniemi conjectured in 1966 that $$\chi(G \times H) = \min\{\chi(G), \chi(H)\}$$ for any graphs G and H. Here $$G\times H$$ is the graph with vertex set $$V(G)\times V(H)$$ defined by putting $$(x,y)$$ and $$(x^{\prime}, y^{\prime})$$ adjacent if and only if $$xx^{\prime}\in E(G)$$ and $$yy^{\prime}\in V(H)$$. This conjecture received a lot of attention in the past half century. It was disproved recently by Shitov. The Poljak-Rodl function is defined as $$f(n) = \min\{\chi(G \times H): \chi(G)=\chi(H)=n\}$$. Hedetniemi's conjecture is equivalent to saying $$f(n)=n$$ for every integer $$n$$. Shitov’s result shows that $$f(n)<n$$ when $$n$$ is sufficiently large. Using Shitov’s result, Tardif and Zhu showed that $$f(n) \le n - (\log n)^{1/4-o(1)}$$ for sufficiently large $$n$$. Using Shitov’s method, He and Wigderson showed that for $$\epsilon \approx 10^{-9}$$ and $$n$$ sufficiently large, $$f(n) \le (1-\epsilon)n$$. In this note we observe that a slight modification of the proof in the paper of Zhu and Tardif shows that $$f(n) \le (\frac 12 + o(1))n$$ for sufficiently large $$n$$. On the other hand, it is unknown whether $$f(n)$$ is bounded by a constant. However, we do know that if $$f(n)$$ is bounded by a constant, then the smallest such constant is at most 9. This note gives self-contained proofs of the above mentioned results.
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The k-independence number of a graph G, denoted as α k (G), is the order of a largest induced k-colorable subgraph of G. In [S. Špacapan, The k-independence number of direct products of graphs, European J. Combin. 32 (2011) 1377–1383] the author conjectured that the direct product G × H of graphs G and H obeys the following bound α k (G × H) ≤ α k (G)|V (H)| + α k (H)|V (G)| − α k (G)α k (H), and proved the conjecture for k = 1 and k = 2. If true for k = 3 the conjecture strenghtens the result of El-Zahar and Sauer who proved that any direct product of 4-chromatic graphs is 4-chromatic [M. El-Zahar and N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985) 121–126]. In this paper we prove that the above bound is true for k = 3 provided that G and H are graphs that have complete tripartite subgraphs of orders α 3 (G) and α 3 (H), respectively.
Article
The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and is NP-complete otherwise. One of the important questions related to the dichotomy conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each nontrivial set of height 1 identities a structure within the range of the conjecture whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of nontrivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of nontrivial height 1 identities differ for ω \omega -categorical structures with less than doubly exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.
Article
Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant δ>0 such that for all c sufficiently large, there exist graphs G and H with chromatic number at least (1+δ)c for which χ(G×H)≤c.
Chapter
This glossary contains an annotated listing of some 300 parameters of graphs, together with their definitions, and, for most of these, a reference to the authors who introduced them. Let G = (V, E) be an undirected graph having order n = |V | vertices and size m = |E| edges. Two graphs G and H are isomorphic, denoted G ≃ H, if there exists a bijection ϕ : V (G) → V (H) such that two vertices u and v are adjacent in G if and only if the two vertices ϕ(u) and ϕ(v) are adjacent in H. For the purposes of this paper, we shall say that a parameter of a graph G is any integer-valued function $$f: \mathcal {G} \rightarrow \mathcal {Z}$$ from the class of all finite graphs $$\mathcal {G}$$ to the integers $$\mathcal {Z}$$, such that for any two graphs G and H, if G is isomorphic to H then f(G) = f(H). This glossary also contains a listing of some 70 conjectures related to these parameters, more than 26 new parameters and open problem areas for study, and some 600 references to papers in which these parameters were introduced and then studied.
Article
The $$\mathbb {Z}_2$$-index $$\mathrm{ind}(X)$$ of a $$\mathbb {Z}_2$$-CW-complex X is the smallest number n such that there is a $$\mathbb {Z}_2$$-map from X to $$S^n$$. Here we consider $$S^n$$ as a $$\mathbb {Z}_2$$-space by the antipodal map. Hedetniemi’s conjecture is a long standing conjecture in graph theory concerning the graph coloring problem of tensor products of finite graphs. We show that if Hedetniemi’s conjecture is true, then $$\mathrm{ind}(X \times Y) = \min \{ \mathrm{ind}(X) , \mathrm{ind}(Y)\}$$ for every pair X and Y of finite $$\mathbb {Z}_2$$-complexes.
Article
Interpretations within Formal Languages and morphisms within the category of graphs are just graph homomorphisms in disguise. Many classic results have been obtained in these contexts. This chapter discusses generalized colorings. Specifically, a homomorphism from a graph G to the r-clique is just an r-coloring of G. In this context, the preimage of any vertex is just a color class. Considering X(G) denoting the chromatic number of G, the chapter discusses that if f:G → H, then X(G) <= X (H). There has been a flurry of activity on graph homomorphisms in the past few years. The chapter discusses homomorphisms and parameters such as the chromatic number and independence. It highlights the obstructions to the existence of homomorphisms and describes the homomorphism order.
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DISSERTATION (PH.D.)--THE UNIVERSITY OF MICHIGAN Dissertation Abstracts International,
Erdos andL. Lovász, On graphs of Ramsey type,Ars Comb
• S A Burr