The second author's (B.A.R.) ω, Δ, χ conjecture proposes that every graph satisfies . In this article, we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way, we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colorable complement. To prove our results, we introduce a very useful χ-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called *skeletal* graphs.

The square *G*^{2} of a graph *G* is the graph defined on such that two vertices *u* and *v* are adjacent in *G*^{2} if the distance between *u* and *v* in *G* is at most 2. Let and be the chromatic number and the list chromatic number of a graph *H*, respectively. A graph *H* is called *chromatic-choosable* if . It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that for every graph *G*, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value can be arbitrarily large.

Let *G* be a bridgeless cubic graph. Consider a list of *k* 1-factors of *G*. Let be the set of edges contained in precisely *i* members of the *k* 1-factors. Let be the smallest over all lists of *k* 1-factors of *G*. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if , then is an upper bound for the girth of *G*. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4-cycle cover of length and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang . We also give a negative answer to a problem stated in .

For a class of graphs *X*, let be the number of graphs with vertex set in the class *X*, also known as the speed of *X*. It is known that in the family of hereditary classes (i.e. those that are closed under taking induced subgraphs) the speeds constitute discrete layers and the first four lower layers are constant, polynomial, exponential, and factorial. For each of these four layers a complete list of minimal classes is available, and this information allows to provide a global structural characterization for the first three of them. The minimal layer for which no such characterization is known is the factorial one. A possible approach to obtaining such a characterization could be through identifying all minimal superfactorial classes. However, no such class is known and possibly no such class exists. To overcome this difficulty, we employ the notion of boundary classes that has been recently introduced to study algorithmic graph problems and reveal the first few boundary classes for the factorial layer.

A *rainbow matching* for (not necessarily distinct) sets of hypergraph edges is a matching consisting of *k* edges, one from each . The aim of the article is twofold—to put order in the multitude of conjectures that relate to this concept (some first presented here), and to prove partial results on one of the central conjectures.

Let *C* be a given circuit of a bridgeless cubic graph *G*. It was conjectured by Seymour that *G* has a circuit double cover (CDC) containing the given circuit *C*. This conjecture (strong CDC [SCDC] conjecture) has been verified by Fleischner and Häggkvist for various families of graphs and circuits. In this article, some of these earlier results have been improved: (1) if contains a Hamilton path or a *Y*-tree of order less than 14, then *G* has a CDC containing *C*; (2) if is connected and , then *G* has a CDC containing *C*.

Asteroidal Triple-free (AT-free) graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component. (AT-free graphs have asteroidal number at most 2.) In this article, we characterize graphs of bounded asteroidal number by means of a vertex elimination ordering, thereby solving a long-standing open question in algorithmic graph theory. Similar characterizations are known for chordal, interval, and cocomparability graphs.

Immersion is a containment relation on graphs that is weaker than topological minor. (Every topological minor of a graph is also its immersion.) The graphs that do not contain any of the Kuratowski graphs (*K*_{5} and *K*_{3, 3}) as topological minors are exactly planar graphs. We give a structural characterization of graphs that exclude the Kuratowski graphs as immersions. We prove that they can be constructed from planar graphs that are subcubic or of branch-width at most 10 by repetitively applying *i*-edge-sums, for . We also use this result to give a structural characterization of graphs that exclude *K*_{3, 3} as an immersion.

Bouchet's conjecture asserts that each signed graph which admits a nowhere-zero flow has a nowhere-zero 6-flow. We verify this conjecture for two basic classes of signed graphs—signed complete and signed complete bipartite graphs by proving that each such flow-admissible graph admits a nowhere-zero 4-flow and we characterise those which have a nowhere-zero 2-flow and a nowhere-zero 3-flow.

Given two graphs *F* and *G*, an *induced* *F*-decomposition of *G* is a partition of into induced subgraphs isomorphic to *F*. Bondy and Szwarcfiter [*J. Graph Theory*, DOI: 10.1002/jgt.21654] defined the value as the maximum number of edges in a graph of order *n* admitting an *induced* *F*-decomposition and determined the value of for some graphs (and families of graphs). In this article, we prove that is valid for all graphs *F*. We also present tighter asymptotic bounds for some of the small graphs for which the exact value of remains unknown. The proofs are based on the heavy use of various classes of Kneser graphs and hypergraphs.

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let denote the number of cops needed to capture the robber in a graph *G* in this variant, and let denote the treewidth of *G*. We show that if *G* is planar then , and there is a polynomial-time constant-factor approximation algorithm for computing . We also determine, up to constant factors, the value of of the Erdős–Rényi random graph for all admissible values of *p*, and show that when the average degree is ω(1), is typically asymptotic to the domination number.

Let *U*_{5} be the tournament with vertices *v*_{1}, …, *v*_{5} such that , and if , and . In this article, we describe the tournaments that do not have *U*_{5} as a subtournament. Specifically, we show that if a tournament *G* is “prime”—that is, if there is no subset , , such that for all , either for all or for all —then *G* is *U*_{5}-free if and only if either *G* is a specific tournament or can be partitioned into sets *X*, *Y*, *Z* such that , , and are transitive. From the prime *U*_{5}-free tournaments we can construct all the *U*_{5}-free tournaments. We use the theorem to show that every *U*_{5}-free tournament with *n* vertices has a transitive subtournament with at least vertices, and that this bound is tight.

A relational structure is (connected-) homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions that generalise (connected-) homogeneity, where ‘isomorphism’ may be replaced by ‘homomorphism’ or ‘monomorphism’ in the definition. In particular, we study the classes of finite connected-homomorphism-homogeneous graphs, with the aim of producing classifications. The main result is a classification of the finite graphs, where a graph *G* is if every homomorphism from a finite connected induced subgraph of *G* into *G* extends to an endomorphism of *G*. The finite (connected-homogeneous) graphs were classified by Gardiner in 1976, and from this we obtain classifications of the finite and finite graphs. Although not all the classes of finite connected-homomorphism-homogeneous graphs are completely characterised, we may still obtain the final hierarchy picture for these classes.

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2-distance -colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2-distance -colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree *m* less than and with large enough maximum degree Δ (depending only on *m*) can be list 2-distance -colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2-distance -colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2-distance -colored (*C* depending only on *m*). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2-distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.

We study the following problem: given a real number *k* and an integer *d*, what is the smallest ε such that any fractional -precoloring of vertices at pairwise distances at least *d* of a fractionally *k*-colorable graph can be extended to a fractional -coloring of the whole graph? The exact values of ε were known for and any *d*. We determine the exact values of ε for if , and if , and give upper bounds for if , and if . Surprisingly, ε viewed as a function of *k* is discontinuous for all those values of *d*.

Deciding whether a digraph contains a pair of arc-disjoint in- and out-branchings rooted at a specified vertex is a well-known NP-complete problem (as proved by Thomassen, see ). This problem has been shown to be polynomial time solvable for semicomplete digraphs and for quasi-transitive digraphs . In this article, we study the problem for locally semicomplete digraphs. We characterize locally semicomplete digraphs that contain a pair of arc-disjoint in- and out-branchings rooted at a specified vertex. Our proofs are constructive and imply the existence of a polynomial time algorithm for finding the desired branchings when they exist. Our results generalizes those from for semicomplete digraphs and solves an open problem from .

In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two-part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that *p* is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .

A *broom* is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph *G* contains a spanning tree that is a broom. In this article, we give one solution to this problem by demonstrating that if *G* is a connected graph of order with , then *G* contains a spanning broom. This result is best possible.

Tutte's 3-Flow Conjecture states that every 2-edge-connected graph with no 3-cuts admits a 3-flow. The 3-Flow Conjecture is equivalent to the following: let *G* be a 2-edge-connected graph, let *S* be a set of at most three vertices of *G*; if every 3-cut of *G* separates *S* then *G* has a 3-flow. We show that minimum counterexamples to the latter statement are 3-connected, cyclically 4-connected, and cyclically 7-edge-connected.

A sequence is *nonrepetitive* if it contains no identical consecutive subsequences. An edge coloring of a path is *nonrepetitive* if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Soták, and Škrabul'áková proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovász Local Lemma). Our approach is based on the Moser–Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmović, Joret, Kozik, and Wood.

The *circular chromatic index* of a graph *G*, written , is the minimum *r* permitting a function such that whenever *e* and are adjacent. It is known that for any , there is a 3-regular simple graph *G* with . This article proves the following results: Assume is an odd integer. For any , there is an *n*-regular simple graph *G* with . For any , there is an *n*-regular multigraph *G* with .

Let *G* be a connected graph of order *n* and independence number α. We prove that *G* has a spanning tree with average distance at most , if , and at most , if . As a corollary, we obtain, for *n* sufficiently large, an asymptotically sharp upper bound on the average distance of *G* in terms of its independence number. This bound, apart from confirming and improving on a conjecture of Graffiti [8], is a strengthening of a theorem of Chung [1], and that of Fajtlowicz and Waller [8], on average distance and independence number of a graph.

An edge-coloring of a graph *G* with colors is called an interval *t*-coloring if all colors are used, and the colors of edges incident to any vertex of *G* are distinct and form an interval of integers. In 1991, Erdős constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

The well-known Ramsey number is the smallest integer *n* such that every -free graph of order *n* contains an independent set of size *u*. In other words, it contains a subset of *u* vertices with no *K*_{2}. Erdős and Rogers introduced a more general problem replacing *K*_{2} by for . Extending the problem of determining Ramsey numbers they defined the numbers

where the minimum is taken over all -free graphs *G* of order *n*. In this note, we study an analogous function for 3-uniform hypergraphs. In particular, we show that there are constants *c*_{1} and *c*_{2} depending only on *s* such that

We present a transformation on a chordal 2-connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers *n*, *m* with , the threshold graph having *n* vertices and *m* edges that consists of an -clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2-connected chordal graphs on *n* vertices and *m* edges.

For a graph *G*, let be the probability that three distinct random vertices span exactly *i* edges. We call the *3-local profile*
of *G*. We investigate the set of all vectors that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of to the plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality . We also give a full description of the triangle-free case, i.e. the intersection of with the hyperplane . This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.