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# Homomorphisms of graphs and automata

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## Abstract

DISSERTATION (PH.D.)--THE UNIVERSITY OF MICHIGAN Dissertation Abstracts International,

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... 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)}. ...
... Applying Lemma 3.6 with n = k+3, we obtain that if one of the assumptions (i) and (ii) of the theorem holds, then χ(G × H) = k. Thus we may assume that k = 5 and (|V (G)|, |V (H)|) ∈ {(8, 8),(8,9), (8, 10),(9,8),(10,8)}. We may assume that |V (G)| = 8. ...
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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.
... A tentative proof was sketched, based on what turned out to be an independent formulation of the conjecture of Hedetniemi [5] on the chromatic number of a categorical product of graphs. In recent years, Shitov [9] has refuted Hedetniemi's conjecture. ...
... This was the first appearance in a journal article of a conjecture that had been formulated ten years earlier by Hedetniemi in the technical report [5]. The conjecture gained popularity in the eighties, with strong partial results being proved while the general case remained seemingly intractable. ...
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We revisit the Burr-Erdős-Lovász conjecture on chromatic Ramsey numbers. We show that it admits a proof based on the Lovász ϑ parameter in addition to the proof of Xuding Zhu based on the fractional chromatic number. However, there are no proofs based on topological lower bounds on chromatic numbers, because the chromatic Ramsey numbers of generalised Mycielski graphs are too large. We show that the 4-chromatic generalised Mycielski graphs other than $K_4$ all have chromatic Ramsey number 14, and that the n-chromatic generalised Mycielski graphs all have chromatic Ramsey number at least $2^{n/4}$ .
... There is a wide literature about the behavior of graph parameters under graph operations, and in particular graph products. For the chromatic number, it includes the famous conjecture of Hedetniemi (1966) [18], that remained open more than fifty years, it was shown to hold for many particular classes, and was recently disproved by Shitov [33]. Other results on the chromatic number and its variations in product graphs can be found in [1,3,6,8,11,13,15,19,20,21,22,23,24,25,26,27,32,34,36], and on domination in product graphs in [16,17,20,21]. ...
... There is a wide literature about the behavior of graph parameters under graph operations, and in particular graph products. For the chromatic number, it includes the famous conjecture of Hedetniemi (1966) [18], that remained open more than fifty years, it was shown to hold for many particular classes, and was recently disproved by Shitov [33]. Other results on the chromatic number and its variations in product graphs can be found in [1,3,6,8,11,13,15,19,20,21,22,23,24,25,26,27,32,34,36], and on domination in product graphs in [16,17,20,21]. ...
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The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.
... Hedetniemi's conjecture [Hed66] for c-colorings states that the tensor product G×H is c-colorable if and only if G or H is c-colorable. El-Zahar & Sauer [ES85] proved it for c = 3. ...
... Shitov's counterexample [Shi19] to Hedetniemi's conjecture [Hed66] relies on the existence of a graph F which on one hand has high odd girth (> 5, meaning no cycles of length 3 nor 5), and on the other hand has high fractional chromatic number (χ f > 3). The second condition means that the chromatic number of the lexicographic product F [K k ] ("blowing-up" each vertex into a k-clique with all possible edges between adjacent cliques) increases with k as χ(F [K k ]) ≥ χ f (F )·k for all k. ...
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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).
... Conjecture 3.1 Let G and H be graphs then χ(G ⊗ H) = min{χ(G), χ(H)}. (Hedetniemi, 1966) Theorem 3.2 If |X 1 | = n ≥ 2 and |X 2 | = m ≥ 3 then ω(Γ 1 ⊗ Γ 2 ) = χ(Γ 1 ⊗ Γ 2 ) = min{n, m}. ...
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\Gamma (SL_{X})$is defined and has been investigated in (Toker, 2016). In this paper our main aim is to extend this study over$\Gamma (SL_{X})$to the tensor product. The diameter, radius, girth, domination number, independence number, clique number, chromatic number and chromatic index of$\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$has been established. Moreover, we have determined when$\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$is a perfect graph. ... A famous conjecture of Hedetniemi ( [12], [17]) states that always equality occurs. We denote by 2K 2 the graph consisting of two disjoint edges. ... Article Full-text available It is shown that the chromatic number X(G) = k of a uniquely colorable Cayley graph G over a group Γ is a divisor of /Γ/ = n. Each color class in a k-coloring of G is a coset of a subgroup of order n=k of Γ. Moreover, it is proved that (k - 1)n is a sharp lower bound for the number of edges of a uniquely k-colorable, noncomplete Cayley graph over an abelian group of order n. Finally, we present constructions of uniquely colorable Cayley graphs by graph products. ... That χ(G × H) ≤ min{χ(G), χ(H)} follows easily. However, the Hedetniemi conjecture (see [7]) states that ... Article Full-text available If distinct colours represent distinct technology types that are placed at the vertices of a simple graph in accordance to a minimum proper colouring, a disaster recovery strategy could rely on an answer to the question: "What is the maximum destruction, if any, the graph (a network) can undergo while ensuring that at least one of each technology type remain, in accordance to a minimum proper colouring of the remaining induced subgraph." In this paper, we introduce the notion of a chromatic core subgraph$H$of a given simple graph$G$in answer to the stated problem. Since for any subgraph$H$of$G$it holds that$\chi(H) \le \chi(G)$, the problem is well defined. ... (For example, see the survey of Zhu [Zhu98].) The conjecture that the first statement is true was formulated by Hedetniemi [Hed66]. Exponential graphs were used by El-Zahar and Sauer to prove the first nontrivial case of Hedetniemi's conjecture: the categorical product of two 4-chromatic graphs is 4-chromatic [ES85]. ... Preprint Let$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$. We present such an algorithm, answering Tardif's question affirmatively. ... When G and H are loopless graphs (Hedetniemi, 1966) conjectured the stronger relation ... Preprint We study the design of stochastic local search methods to prove unsatisfiability of a constraint satisfaction problem (CSP). For a binary CSP, such methods have been designed using the microstructure of the CSP. Here, we develop a method to decompose the microstructure into graph tensors. We show how to use the tensor decomposition to compute a proof of unsatisfiability efficiently and in parallel. We also offer substantial empirical evidence that our approach improves the praxis. For instance, one decomposition yields proofs of unsatisfiability in half the time without sacrificing the quality. Another decomposition is twenty times faster and effective three-tenths of the times compared to the prior method. Our method is applicable to arbitrary CSPs using the well known dual and hidden variable transformations from an arbitrary CSP to a binary CSP. ... Conjecture 1.1. [15] For any two graphs G and H, χ(G × H) = min{χ(G), χ(H)}. ... Preprint The neighborhood complex$\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. LetM(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$. Therefore, we enrich the family of graphs satisfying the Hedetniemi's conjecture. ... On the other hand, Bonsma and Cereceda [2] showed that the complexity jumps drastically for larger cliques: for every fixed k ≥ 4, the K k -Recolouring problem is PSPACE-complete. 1 Later, Brewster, McGuinness, Moore and Noel [5] extended this dichotomy to the case when H is a "circular clique. " Wrochna [21] developed ideas inspired by algebraic topology to prove the remarkably general result that H-Recolouring is solvable in polynomial time whenever H does not contain a cycle of length 4. By further refining his topological approach, Wrochna [22] (see also [23]) proved a "multiplicativity" result for graphs without cycles of length 4 which is closely connected to Hedetniemi's Conjecture [11]; very recently, Tardif and Wrochna [19] have extended these methods beyond the setting of C 4 -free graphs. On the hardness side, however, only a few examples are known. ... Preprint Full-text available Given a loop-free graph$H$, the reconfiguration problem for homomorphisms to$H$(also called$H$-colourings) asks: given two$H$-colourings$f$of$g$of a graph$G$, is it possible to transform$f$into$g$by a sequence of single-vertex colour changes such that every intermediate mapping is an$H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs (e.g. all$C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever$H$is a$K_{2,3}$-free quadrangulation of the$2$-sphere (equivalently, the plane) which is not a$4$-cycle. If we instead consider graphs$G$and$H$with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for$H$-colourings is PSPACE-complete whenever$H$is a reflexive$K_{4}$-free triangulation of the$2$-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which$H$-Recolouring is known to be PSPACE-complete for reflexive instances. ... So χ(G × H) ≤ χ(G), and similarly χ(G × H) ≤ χ(H). 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. ... Preprint 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 all finite graphs G and H. 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]. ... 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$\chi(G \times H) \le c$. ... Conjecture 1.1 ([15]). If G and H are graphs, then ... Article Full-text available 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. ... Therefore f (n) ≤ n. 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]. ... Preprint 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$\lim_{n \to \infty} \frac{f(n)}{n} \le \frac 12$. ... Conjecture 1 (Hedetniemi [10]) Let G, H be graphs. Then, ... Article Given a proper coloring$f$of a graph$G$, a b-vertex in$f$is a vertex that is adjacent to every color class but its own. It is a b-coloring if every color class contains at least one b-vertex, and it is a fall-coloring if every vertex is a b-vertex. The b-chromatic number of$G$is the maximum integer$b(G)$for which$G$has a b-coloring with$b(G)$colors, while the fall-chromatic number and the fall-acromatic number of$G$are, respectively, the minimum and maximum integers$\chi_f(G),\psi_f(G)$for which$G$has a fall-coloring. In this article, we explore the concepts of b-homomorphisms and Type II homomorphisms, which generalize the concepts of b-colorings and fall-colorings, and present some meta-theorems concerning products of graphs. In particular, our results give new lower and upper bounds for these metrics on the main existing graph products. We also give some results about the chromatic number of the direct product$G\times H$, and as a consequence we get that the Hedetniemi's Conjecture holds whenever$G$or$H$is a$2K_2$-free graph, or a perfect graph. Finally, we give a negative answer to a question posed by Kaul and Mitillos about fall-colorings of perfect graphs. ... So χ(G × H) χ(G), and similarly χ(G×H) χ(H). Hedetniemi conjectured in 1966 that χ(G×H) = min{χ(G), χ(H)} for all finite graphs G and H [6]. The conjecture received a lot of attention [7,10,13,14] and remained open for more than half century. ... 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$\min\{\chi(G), \chi(H)\}$. ... The classical conjecture of S. T. Hedetniemi [8] posited the equality for all G and H. More than 50 years have passed since the conjecture appeared, and it keeps attracting serious attention of researchers working in graph theory and combinatorics; we mention four exhaustive survey papers [9,11,13,18] for more detailed information on the topic. ... Preprint Full-text available The chromatic number of$G\times H$can be smaller than the minimum of the chromatic numbers of finite simple graphs$G$and$H$. ... Therefore χ(G × H) ≤ min{χ(G), χ(H)}. 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]). ... 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. ... The question of whether, given a graph parameter w and a graph product ⋆, w(G ⋆ H) is bounded by a function of w(G) and w(H) is natural and has been studied many times. One example is Hedetniemi's conjecture [22], posed in 1966, which states that for the chromatic number χ, the tensor product ×, and any graphs G, H, we have χ(G × H) = min{χ(G), χ(H)}. This conjecture was proved false by Shitov [40] in 2019, who described a pair of graphs G and H satisfying χ(G × H) < min{χ(G), χ(H)}. ... Preprint Full-text available Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs$G$and$H$and a graph product$\star$, we address the question: is the twin-width of$G\star H$bounded by a function of the twin-widths of$G$and$H$and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, rooted, replacement, and zig-zag products. For the lexicographical product we prove that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs. In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs. ... Disproving the so-called Hedetniemi's conjecture [23], it has recently proven [24] that the upper bound described in Lemma 2 may not be reached. In any case, the following result is also known. ... Article Full-text available In this paper, we determine explicitly the r-dynamic chromatic number of the direct product of any given path with either a path or a cycle. Illustrative examples are shown for each one of the cases that are studied throughout the paper. ... The categorical product of two graphs G and H is the graph G × H with vertexset V (G × H) = V (G) × V (H), whose edges are the pairs {(g 1 , h 1 ), (g 2 , h 2 )} such that {g 1 , g 2 } is an edge of G and {h 1 , h 2 } is an edge of H. Shitov [8] proved that the chromatic number of a categorical product of graphs can be smaller than the minimum of the chromatic numbers of the factors, hence disproving Hedetniemi's conjecture of 1966 [6]. ... ... For example, it is well known that the chromatic number of a Cartesian graph (see Section 3 for a formal definition) is the minimum of the chromatic numbers of the components. For tensor graphs, it was conjectured by Hedetniemi [6] that the same result would be true. However, Hedetniemi's conjecture was recently disproved by Shitov [10]. ... Preprint In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two graphs have a perfect$k$-orthogonal colouring, then so does their tensor graph. This provides an upper bound on the$k$-orthogonal chromatic number for general tensor graphs. Lastly, two other conditions for a tensor graph to have a perfect$k\$-orthogonal colouring are given.
... The categorical product of two graphs G and H is the graph G × H with vertexset V (G × H) = V (G) × V (H), whose edges are the pairs {(g 1 , h 1 ), (g 2 , h 2 )} such that {g 1 , g 2 } is an edge of G and {h 1 , h 2 } is an edge of H. Shitov [10] proved that the chromatic number of a categorical product of graphs can be smaller than the minimum of the chromatic numbers of the factors, disproving Hedetniemi's conjecture of 1966 [8]. ...
... On the other hand, Bonsma and Cereceda [2] showed that the complexity jumps drastically for larger cliques: for every fixed k ≥ 4, the K k -Recolouring problem is PSPACEcomplete. 1 Later, Brewster, McGuinness, Moore and Noel [5] extended this dichotomy to the case when H is a "circular clique. " Wrochna [26] developed ideas inspired by algebraic topology to prove the remarkably general result that H-Recolouring is solvable in polynomial time whenever H does not contain a cycle of length 4. By further refining his topological approach, Wrochna [27] (see also [28]) proved a "multiplicativity" result for graphs without cycles of length 4 which is closely connected to Hedetniemi's Conjecture [12]; very recently, Tardif and Wrochna [24] have extended these methods beyond the setting of C 4 -free graphs. ...
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Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs H (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle. From this result, we deduce an analogous statement for non-bipartite K2,3-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and 4-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which the reconfiguration problem is known to be PSPACE-complete for reflexive instances.
... The categorical product of two graphs G and H is the graph G × H with vertexset V (G × H) = V (G) × V (H), whose edges are the pairs {(g 1 , h 1 ), (g 2 , h 2 )} such that {g 1 , g 2 } is an edge of G and {h 1 , h 2 } is an edge of H. Hedetniemi's conjecture of 1966 [12] states that the chromatic number of a categorical product of graphs is equal to the minimum of the chromatic numbers of the factors. In 2019, Shitov [14] refuted the conjecture by constructing counterexamples for very large chromatic numbers. ...
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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.
... Therefore χ(G × H) ≤ min{χ(G), χ(H)}. In 1966, Hedetniemi conjectured in [8] that equality always holds in the above inequality. ...
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.
... Hedetniemi [31] conjectured that for all graphs G and G ′ , χ(G×G ′ ) = min(χ(G), χ(G ′ )). In 2005, Cordova et al. [23] proved this conjecture for all Γ(Z n ), χ(Γ(Z n ) × ****************************************************************************** Surveys in Mathematics and its Applications 15 (2020), 371 -397 http://www.utgjiu.ro/math/sma ...
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This article gives a comprehensive survey on zero-divisor graphs of finite commutative rings. We investigate the results on structural properties of these graphs.
... This proves some cases of Hedetniemi's conjecture of 1966 [6], which states that (2) χ(G × H) = min{χ(G), χ(H)}. ...
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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.
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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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Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
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The connectivity of a graph is a “measure” of its connectedness. Some connected graphs are connected rather “loosely” in the sense that the deletion of a vertex or an edge from the graph destroys the connectedness of the graph. There are graphs at the other extreme as well, such as the complete graphs K n , n ≥ 2, which remain connected after the removal of any k vertices, 1≤ k≤n − 1.
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Graphs serve as mathematical models to analyze many concrete real-world problems successfully. Certain problems in physics, chemistry, communication science, computer technology, genetics, psychology, sociology, and linguistics can be formulated as problems in graph theory. Also, many branches of mathematics, such as group theory, matrix theory, probability, and topology, have close connections with graph theory.
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Directed graphs arise in a natural way in many applications of graph theory. The street map of a city, an abstract representation of computer programs, and network flows can be represented only by directed graphs rather than by graphs. Directed graphs are also used in the study of sequential machines and system analysis in control theory.
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The study of planar and nonplanar graphs and, in particular, the several attempts to solve the four-color conjecture have contributed a great deal to the growth of graph theory. Actually, these efforts have been instrumental to the development of algebraic, topological, and computational techniques in graph theory.
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In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. The set of eigenvalues of a graph G is known as the spectrum of G and denoted by Sp(G). We compute the spectra of some well-known families of graphs—the family of complete graphs, the family of cycles etc. We present Sachs’ theorem on the spectrum of the line graph of a regular graph. We also obtain the spectra of product graphs—Cartesian product, direct product, and strong product. We introduce Cayley graphs and Ramanujan graphs and highlight their importance. Finally, as an application of graph spectra to chemistry, we discuss the “energy of a graph”—a graph invariant that is widely studied these days. All graphs considered in this chapter are finite, undirected, and simple.
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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 present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.
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This chapter presents a collection of theorems in combinatorics, proved in the twenty-first century, which are at the same time great and easy to understand. The chapter is written for undergraduate and graduate students interested in combinatorics, as well as for mathematicians working in other areas of mathematics, who would like to learn about recent achievements in combinatorics without going into technical details.
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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
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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
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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.
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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.
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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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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
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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In a proper k-coloring of a k-chromatic graph, for every two distinct colors there are always adjacent vertices with these colors. This observation has led to a coloring called a complete coloring, which is the primary topic of this chapter.
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Graph theory would not be what it is today if there had been no coloring problems. In fact, a major portion of the 20th-century research in graph theory has its origin in the four-color problem. (See Chap. 8 for details.)
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