Robert Šámal

Charles University in Prague | CUNI · Computer Science Institute of Charles University

We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost‐exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lowe...

In 1983, Bouchet proved that every bidirected graph with a nowhere-zero integer-flow has a nowhere-zero 216-flow, and conjectured that 216 could be replaced with 6. This paper shows that for cyclically 5-edge-connected bidirected graphs that number can be replaced with 8.

Betweenness centrality is a network centrality measure based on the amount of shortest paths passing through a given vertex. A graph is betweenness-uniform (BUG) if all vertices have an equal value of betweenness centrality. In this contribution, we focus on betweenness-uniform graphs with betweenness centrality below one. We disprove a conjecture...

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes -- those of a bouquet of $n$ loops and...

Random 2-cell embeddings of a given graph G G are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, E [ F G ] \mathbb {E}[F_G] , of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the...

Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). J\'anos Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain...

We show that for any sufficiently large graph G avoiding Kk as a minor, we can map vertices v∈V(G) to intervals I(v)⊆[0,1] so that (1) I(u)∩I(v)≠∅ for each edge uv (2) the sum of the squares of the lengths of these intervals is O(k6logk), and (3) the average distance between the intervals is at least 1/25. Balanced separators of G of sublinear size...

By using permutation representations of maps, one obtains a bijection between all maps whose underlying graph is isomorphic to a graph G and products of permutations of given cycle types. By using statistics on cycle distributions in products of permutations, one can derive information on the set of all 2-cell embeddings of G. In this paper, we stu...

Several recent results and conjectures study counting versions of classical existence statements. We ask the same question for circuit double covers of cubic graphs. We prove an exponential bound for planar graphs: Every bridgeless cubic planar graph with n vertices has at least (5/2)n/4-1/2 circuit double covers. The method we used to obtain this...

By using permutation representations of maps, one obtains a bijection between all maps whose underlying graph is isomorphic to a graph $G$ and products of permutations of given cycle types. By using statistics on cycle distributions in products of permutations, one can derive information on the set of all $2$-cell embeddings of $G$. In this paper,...

Two well-known results in the world of nowhere-zero flows are Jaeger's 4-flow theorem asserting that every 4-edge-connected graph has a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow and Seymour's 6-flow theorem asserting that every 2-edge-connected graph has a nowhere-zero $\mathbb{Z}_6$-flow. Dvo\v{r}\'ak and the last two authors of this pa...

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from a Cayley graph $\mathrm{Cay}(M, B)$ to another Cayley graph $\mathrm{Cay}(M', B')$ then every graph with an $(M,B)$-flow has an $(M',B')$-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conj...

Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, wh...

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of ℤ3-, ℤ4-, and ℤ6-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen.

We answer a question on group connectivity suggested by Jaeger et al [J. Combin. Theory, Ser. B 56 (1992), pp. 165–182]: we find that Z 2 2‐connectivity does not imply Z 4‐connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify their properties using a nontrivial enumerative algorithm (and we also use an...

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from Cayley graph Cay(M, B) to another Cayley graph Cay(M', B') then every graph with an (M, B)-flow has an (M', B')-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in al...

Mantel's Theorem asserts that a simple $n$ vertex graph with more than $\frac{1}{4}n^2$ edges has a triangle (three mutually adjacent vertices). Here we consider a rainbow variant of this problem. We prove that whenever $G_1, G_2, G_3$ are simple graphs on a common set of $n$ vertices and $|E(G_i)| > ( \frac{ 26 - 2 \sqrt{7} }{81})n^2 \approx 0.255...

Given η ∈ [0, 1], a colouring C of V (G) is an η-majority colouring if at most ηd⁺ (v) out-neighbours of v have colour C(v), for any v ∈ V (G). We show that every digraph G equipped with an assignment of lists L, each of size at least k, has a 2/k-majority L-colouring. For even k this is best possible, while for odd k the constant 2/k cannot be rep...

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...

We answer a question on group connectivity suggested by Jaeger et al. [Group connectivity of graphs -- A nonhomogeneous analogue of nowhere-zero flow properties, JCTB 1992]: we find that $\mathbb Z_2^2$-connectivity does not imply $\mathbb Z_4$-connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify thei...

An embedding $i \mapsto p_i\in \mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\in V(G)$. The...

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of $Z_3$-, $Z_4$-, and $Z_6$-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen...

We introduce a nonlocal game that captures and extends the notion of graph isomorphism. This game can be won in the classical case if and only if the two input graphs are isomorphic. Thus, by considering quantum strategies we are able to define the notion of quantum isomorphism. We also consider the case of more general non-signalling strategies, a...

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from Cayley graph Cay(M, B) to another Cayley graph Cay(M′, B′) then every graph with (M, B)-flow has (M′, B′)-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all case...

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. Our results may be seen as a counting alternative to the well-known proofs of existence of -, -, and -flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen.

We prove that every 3-edge-connected graph $G$ has a 3-flow $\phi$ with the property that $|\mathop{supp}(\phi)| \ge \frac{5}{6} |E(G)|$. The graph $K_4$ demonstrates that this $\frac{5}{6}$ ratio is best possible; there is an infinite family where $\frac 56$ is tight.

Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a fin...

Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists $L$, each of size at least $k$, has a $2/k$-majority $L$-colouring. For even $k$ this is best possible, while...

We introduce a two-player nonlocal game, called the $(G,H)$-isomorphism game, where classical players can win with certainty if and only if the graphs $G$ and $H$ are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the $(G,H)$-isomorphism game, respec...

A vector $t$-coloring of a graph $G$ is an assignment of unit vectors $i\mapsto p_i$ to its vertices such that $\langle p_i, p_j\rangle\le -1/(t-1),$ for all $i\sim j$. The vector chromatic number of $G$, denoted by $\chi_v(G)$, is the smallest real number $t \ge 2$ for which a vector $t$-coloring exists. We use vector colorings to study graph homo...

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this definition is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph, which, if solved positively, would imply several other important conjectures (e.g., t...

Tutte's famous 5-flow conjecture asserts that every bridgeless graph has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. Here we give (two versions of) a new proof of Seymour's Theorem. Both are roughly equal to Seymour's in terms of complexity, but they offer an alternative perspective which we hope will be o...

We study drawings of graphs on the torus with crossings allowed. A question posed in [M. DeVos, B. Mohar, and R. Šámal, Unexpected behaviour of crossing sequences, J. Combin. Theory Ser. B 101 (2011), no. 6, 448–463], specialized to the case of the torus, asks, whether for every disconnected graph there is a drawing in the torus with the minimal nu...

We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the fact...

The star chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree $\Delta=\Delta(G)$. Our best lower bound on $\chi_s'$ in terms of $\Delta$ is $2\Delta(1+o(1))$...

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph. Answering a question of DeVos, Ne\v{s}et\v{r}il, and Raspaud, we prove that there exists...

Many questions at the core of graph theory can be formulated as questions about certain group-valued flows: examples are the cycle double cover conjecture, Berge-Fulkerson conjecture, and Tutte's 3-flow, 4-flow, and 5-flow conjectures. As an approach to these problems Jaeger and DeVos, Ne\v{s}et\v{r}il, and Raspaud define a notion of graph morphism...

The guarding game is a game in which several cops try to guard a region in a (directed or undirected) graph against Robber. Robber and the cops are placed on the vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, Robber on the remaining vertices (the robber-region). The goal of Robber...

Petersen coloring (defined by F. Jaeger [Ann. Discrete Math. 8, 123–126 (1980; Zbl 0448.05023)]) is a mapping from the edges of a cubic graph to the edges of the Petersen graph, so that three edges adjacent at a vertex are mapped to three edges adjacent at a vertex. The existence of such mapping for every cubic bridgeless graph is known to imply th...

We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each `bui...

The guarding game is a game in which several cops has to guard a region in a (directed or undirected) graph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, the robber on the remaining vertices (the robber-region). The goal of...

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [J Graph Theory 6(2) (1982), 115–121] and...

The guarding game is a game in which several cops try to guard a region in a (directed or undirected) graph against a robber. The robber and the cops are placed on the vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, the robber on the remaining vertices (the robber-region). The goal...

The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.400m. We show that every cubic bridgeless graph has a cycle cover of total length at most 34m/21 � 1.619m and every bridgeless graph with minimum degree three has a cycle cover...

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of...

The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar. Co...

Eberhard proved that for every sequence $(p_k), 3\le k\le r, k\ne 5,7$ of non-negative integers satisfying Euler's formula $\sum_{k\ge3} (6-k) p_k = 12$, there are infinitely many values $p_6$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge3$, where $p_k=0$ if $k>r$. In this paper we prov...

The nth crossing number of a graph G, denoted cr n (G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr 0 (G)=a, cr 1 (G)=b, and cr 2 (G)=0. This provides support for a conjecture of Archdeacon et al. [D. Archdeacon, C.P. Bonnington, and J....

Tension-continuous (shortly ) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. At the same time, tension-continuous mappings are a dual notion to flow-continuous mappings, and the context of nowhere-zero flows motivates several questions considered in this paper.Extending our earlier research we define new...

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n) vertices, where c is a positive constant. The best known upper bound is (2 + o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “(3,6)-fullerenes,” have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form {λ,−λ} except for the four eigenvalues {3,−1,−1,−1...

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(clogn) vertices, where c is a positive constant. The best known upper bound is (2+o(1))n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}....

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interested in its own, it is useful for study of cut-continuous mappings (defined in [DeVos, M., J. Nešetřil and A. Raspaud, On flow and tension-continuous maps, KAM-DIMATIA Series 567 (2002)]) and homomorphism. Connections with fractional chromat...

We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly TT mappings). Existence of a TT mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see [Hell-Nesetril]). In this p...

We prove that the strong product of any n connected graphs of maximum degree at most n contains a Hamilton cycle. In particular, GΔ(G) is hamiltonian for each connected graph G, which answers in affirmative a conjecture of Bermond, Germa, and Heydemann. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 299–321, 2005 Supported by KONTAKT ME 337 as a...

We prove that the strong product of graphs G 1×⋯×G n is pancyclic, in particular hamiltonian, for n≈cΔ for any cln(25/12)+1/64≈0.75 whenever all G i are connected graphs with the maximum degree at most Δ.

We present an overview of the theory of nowhere zero flows, in particular the duality of flows and colorings, and the extension to antiflows and strong oriented colorings. As the main result, we find the asymptotic relation between oriented and strong oriented chromatic number. A preliminary version of this paper appears as [10]. The paper is organ...

J. Nešetřil and A. Raspaud [Ann. Inst. Fourier 49, 1037-1056 (1999; Zbl 0921.05034)] defined antisymmetric flow, which is a variant of nowhere zero flow, and a dual notion to strong oriented coloring. We give an upper bound on the number of colors needed for a strong oriented coloring of a planar graph, and hereby we find a small antisymmetric flow...

We study the maximum possible number f(k, l) of intersections of the bound- aries of a simple k-gon with a simple l-gon in the plane for k, l ≥ 3. To determine the number f(k, l) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k ≤ l) the easy upper bound kl − l to kl − ⌈k/6⌉ − l and obtain ex...

The checker number is an invariant of a graph defined as the result of a game played on its vertices. Studying its behavior on the powers of a graph, we found that it is surprisingly similar to the well-known Shannon capacity and we defined an analogous quantity — the limit checker number. Finally, we proved that this number can be calculated as a...

J. J. Charatonik [Commentat. Math. Univ. Carol. 32, No. 2, 377-382 (1991; Zbl 0755.54017)] asked if each hereditarily unicoherent continuum X having the property that each of its proper subcontinua is HU-terminal if and only if it is absolutely terminal, must be atriodic. The authors answer this question in the negative by an example.

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (pos-sibly improperly) so that the complement of each color class is bipar-tite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [8] and Bondy and Locke [2] proved that...

We present an overview of the theory of nowhere zero ows, inparticular the duality of ows and colorings, and the extension to antiows and strong oriented colorings. As the main result, wend theasymptotic relation between oriented and strong oriented chromaticnumber.