*G*is a complete graph or a cycle with an odd number of vertices, or

Let *G* be a graph on *n* vertices, with maximal degree *d*, and not containing as an induced subgraph. We prove:

- 1.
- 2.

Here is the maximal eigenvalue of the Laplacian of *G*, is the independence complex of *G*, and denotes the topological connectivity of a complex plus 2. These results provide improved bounds for the existence of independent transversals in -free graphs.

The Four Color Theorem asserts that the vertices of every plane graph can be properly colored with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly colored with the numbers 1, …, 4 in such a way that every face contains a unique vertex colored with the maximal color appearing on that face. They proved that every plane graph has such a coloring with the numbers 1, …, 6. We prove that every plane graph has such a coloring with the numbers 1, …, 5 and we also prove the list variant of the statement for lists of sizes seven.

An edge (vertex) colored graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colors. Rainbow edge (vertex) connectivity of a graph *G* is the smallest number of colors needed for a rainbow edge (vertex) coloring of *G*. In this article, we propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several known results, as well as some new ones.

Let *G* be a simple undirected connected graph on *n* vertices with maximum degree Δ. Brooks' Theorem states that *G* has a proper Δ-coloring unless *G* is a complete graph, or a cycle with an odd number of vertices. To recolor *G* is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any *k*-coloring, , a Δ-coloring of *G* can be obtained by a sequence of recolorings using only the original *k* colors unless

- –
*G*is a complete graph or a cycle with an odd number of vertices, or - –,
*G*is Δ-regular and, for each vertex*v*in*G*, no two neighbors of*v*are colored alike.

We use this result to study the reconfiguration graph of the *k*-colorings of *G*. The vertex set of is the set of all possible *k*-colorings of *G* and two colorings are adjacent if they differ on exactly one vertex. We prove that for , consists of isolated vertices and at most one further component that has diameter . This result enables us to complete both a structural and an algorithmic characterization for reconfigurations of colorings of graphs of bounded maximum degree.

A coloring of the edges of a graph *G* is strong if each color class is an induced matching of *G*. The strong chromatic index of *G*, denoted by , is the least number of colors in a strong edge coloring of *G*. Chang and Narayanan (J Graph Theory 73(2) (2013), 119–126) proved recently that for a 2-degenerate graph *G*. They also conjectured that for any *k*-degenerate graph *G* there is a linear bound , where *c* is an absolute constant. This conjecture is confirmed by the following three papers: in (G. Yu, Graphs Combin 31 (2015), 1815–1818), Yu showed that . In (M. Debski, J. Grytczuk, M. Sleszynska-Nowak, Inf Process Lett 115(2) (2015), 326–330), Dȩbski, Grytczuk, and Śleszyńska-Nowak showed that . In (T. Wang, Discrete Math 330(6) (2014), 17–19), Wang proved that . If *G* is a partial *k*-tree, in (M. Debski, J. Grytczuk, M. Sleszynska-Nowak, Inf Process Lett 115(2) (2015), 326–330), it is proven that . Let be the line graph of a graph *G*, and let be the square of the line graph . Then . We prove that if a graph *G* has an orientation with maximum out-degree *k*, then has coloring number at most . If *G* is a *k*-tree, then has coloring number at most . As a consequence, a graph with has , and a *k*-tree *G* has .

We show that for every even integer there is *n*_{0} such that, if *H* is a 3-uniform hypergraph on , vertices such that the minimum co-degree of *H* is at least , then *H* can be tiled with copies of a loose cycle on *s* vertices. The co-degree condition is tight.

Let *D* be a finite digraph, and let be nonempty subsets of . The (strong form of) Edmonds' branching theorem states that there are pairwise edge-disjoint spanning branchings in *D* such that the root set of is if and only if for all the number of ingoing edges of *X* is greater than or equal to the number of sets disjoint from *X*. As was shown by R. Aharoni and C. Thomassen (J Graph Theory 13 (1989), 71–74), this theorem does not remain true for infinite digraphs. Thomassen also proved that for the class of digraphs without backward-infinite paths, the above theorem of Edmonds remains true. Our main result is that for digraphs without forward-infinite paths, Edmonds' branching theorem remains true as well.

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every *n*-vertex graph has at most maximal induced matchings, and this bound is the best possible. We prove that every *n*-vertex triangle-free graph has at most maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph *K*_{3, 3}. Our result implies that all maximal induced matchings in an *n*-vertex triangle-free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.

A set *S* of vertices in a hypergraph *H* is strongly independent if no two vertices in *S* belong to a common edge. The strong independence number of *H*, denoted , is the maximum cardinality of a strongly independent set in *H*. The rank of *H* is the size of a largest edge in *H*. The hypergraph *H* is *k*-uniform if every edge of *H* has size *k*. The transversal number, denoted , of *H* is the minimum number of vertices that intersect every edge. Our main result is that for all , the strong independence ratio of a hypergraph *H* with rank *k* and maximum degree 3 satisfies and this bound is achieved for all . In particular, this bound is achieved for the Fano plane. As an application of our result, we show that if *H* is a *k*-uniform hypergraph on *n* vertices with *m* edges and with maximum degree 3 and vertices of degree 3, then . This improves a result due to Chvátal and McDiarmid [Combinatorica 12 (1992), 19–26] who proved that in the case when is even and *H* has maximum degree 3.

If *T* is an *n*-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where *T* is assumed to have cyclic triples for some and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing *T* is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of 4-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in *T*, how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.

We consider the Erdős–Rényi random directed graph process, which is a stochastic process that starts with *n* vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let be a graph with *m* edges obtained after *m* steps of this process. Each edge () of independently chooses a color, taken uniformly at random from a given set of colors. We stop the process prematurely at time *M* when the following two events hold: has at most one vertex that has in-degree zero and there are at least distinct colors introduced ( if at the time when all edges are present there are still less than colors introduced; however, this does not happen asymptotically almost surely). The question addressed in this article is whether has a rainbow arborescence (i.e. a directed, rooted tree on *n* vertices in which all edges point away from the root and all the edges are different colors). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is “yes.”

We study the following independent set reconfiguration problem, called *TAR-Reachability*:
given two independent sets *I* and *J* of a graph *G*, both of size at least *k*, is it possible to transform *I* into *J* by adding and removing vertices one-by-one, while maintaining an independent set of size at least *k* throughout? This problem is known to be PSPACE-hard in general. For the case that *G* is a cograph on *n* vertices, we show that it can be solved in time , and that the length of a shortest reconfiguration sequence from *I* to *J* is bounded by (if it exists). More generally, we show that if is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in using disjoint union and complete join operations. Chordal graphs and claw-free graphs are given as examples of such a class .

For each surface Σ, we define max *G* is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .

We present several general results about drawings of , as a beginning to trying to determine its crossing number. As application, we give a complete proof that the crossing number of *K*_{9} is 36 and that all drawings in one large, natural class of drawings of *K*_{11} have at least 100 crossings.

We study the topic of “extremal” planar graphs, defining to be the maximum number of edges possible in a planar graph on *n* vertices that does not contain a given graph *H* as a subgraph. In particular, we examine the case when *H* is a small cycle, obtaining for all and for all , and showing that both of these bounds are tight.

A graph *H* is *strongly immersed* in *G* if *H* is obtained from *G* by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of *H* are mapped to distinct vertices of *G* (*branch vertices*) and edges of *H* are mapped to pairwise edge-disjoint paths in *G*, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree-like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path-like linear decomposition isolating the high degree vertices.

In this article, we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph. Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a 4-regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.

DeVos and Mohar proved a rough structure theorem about small separations in vertex-transitive graphs [5, 6]. By using a new version of Varopolous isoperimetric inequality, we give an improvement of their lower bound on the expansion in the case of infinite vertex-transitive graphs. Specifically, let be a locally finite connected graph such that there is a group *G* acting discretely and transitively on *X*. If is nonempty and finite such that is connected and , then *X* has a ring-like structure. Moreover, we give a similar asymptotic result under the assumption that *X* is an infinite vertex transitive graph. In the setting of finite groups, we use local expansion to show the existence of a nontrivial cyclic subgroup with an effectively bounded index. Finally, we prove that for any there is a graph *T* and a subgraph *A* of *T*, such that .

In this article, we investigate hamiltonian cycles in plane triangulations. The aim of the article is to find the strongest possible form of Whitney's theorem about hamiltonian triangulations in terms of the decomposition tree defined by separating triangles. We will decide on the existence of nonhamiltonian triangulations with given decomposition trees for all trees except trees with exactly one vertex with degree and all other degrees at most 3. For these cases, we show that it is sufficient to decide on the existence of nonhamiltonian triangulations with decomposition tree *K*_{1, 4} or *K*_{1, 5}. We also give computational results on the size of a possible minimal nonhamiltonian triangulation with these decomposition trees.

We present two main results: a 2-page and a rectilinear drawing of the *n*-dimensional cube . Both drawings have the same number of crossings, even though they are given by different constructions. The first improves the current best general 2-page drawing, while the second is the first nontrivial rectilinear drawing of .

An *internal partition* of an *n*-vertex graph is a partition of *V* such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every *d*-regular graph with vertices has an internal partition. Here we prove this for . The case is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group *G*, we define the two-sided group digraph to have vertex set *G*, and an arc from *x* to *y* if and only if for some and . In common with Cayley graphs and digraphs, two-sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on *L* and *R* under which may be viewed as a simple graph of valency , and we call such graphs two-sided group graphs. We also give sufficient conditions for two-sided group digraphs to be connected, vertex-transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley–Wilf limits. We investigate the maximum number of edges in -interval minor-free bipartite graphs. We determine exact values when and describe the extremal graphs. For , lower and upper bounds are given and the structure of -interval minor-free graphs is studied.

A graph is antimagic if there is a one-to-one correspondence such that for any two vertices , . It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree at least three are antimagic. Whether all nonbipartite regular graphs of even degree are antimagic remained an open problem. In this article, we solve this problem and prove that all even degree regular graphs are antimagic.

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4-vertex-coloring: a plane triangulation *G* has a proper 4-vertex-coloring if and only if the dual of *G* has a proper 3-edge-coloring. A cyclic coloring of a map *G* on a surface *F*^{2} is a vertex-coloring of *G* such that any two vertices *x* and *y* receive different colors if *x* and *y* are incident with a common face of *G*. In this article, we extend the result by Tait to two directions, that is, considering maps on a nonspherical surface and cyclic 4-colorings.

The circumference of a graph *G* is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5-, and 6-edge-connected cubic graphs with circumference ratio bounded from above by 0.876, 0.960, and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 4-edge-connected cubic graph is at least 0.75. Up to our knowledge, no upper bounds on this ratio have been known before for cubic graphs with cyclic edge-connectivity above 3. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hägglund and K. Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc Math 312 (2012), 2540–2544].

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph . We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo *q*. We determine the asymptotic distribution of all fixed two-component graphs in for all *q*, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all *q*. We also prove a negative result that no recursive proof of the simplest form exists for a uniform asymptotic distribution for arbitrary graphs.

Let *G* be a Class 1 graph with maximum degree 4 and let be an integer. We show that any proper *t*-edge coloring of *G* can be transformed to any proper 4-edge coloring of *G* using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent.

We determine the 2-color Ramsey number of a *connected*
triangle matching that is any connected graph containing *n* vertex disjoint triangles. We obtain that , somewhat larger than in the classical result of Burr, Erdős, and Spencer for a triangle matching, . The motivation is to determine the Ramsey number of the square of a cycle . We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an “almost” square of a cycle (a cycle of length *n* in which all but at most a constant number *c* of short diagonals are present) is asymptotic to .

A graph is *periodic* if it can be obtained by joining identical pieces in a cyclic fashion. It is shown that the limit crossing number of a periodic graph is computable. This answers a question of Richter [1, Problem 4.2].

Let be a function on the vertex set of the graph . The graph *G* is *f*-*choosable* if for every collection of lists with list sizes specified by *f* there is a proper coloring using colors from the lists. The sum choice number, , is the minimum of , over all functions *f* such that *G* is *f*-choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1–33, Random Struct Algor 16 (2000), 364–368) that if *G* has average degree *d*, then the usual choice number is at least , so they grow simultaneously. In this article, we show that can be bounded while the minimum degree . Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph .

Projective planar graphs can be characterized by a set of 35 excluded minors. However, these 35 are not equally important. A set of 3-connected members of is *excludable* if there are only finitely many 3-connected nonprojective planar graphs that do not contain any graph in as a minor. In this article, we show that there are precisely two minimal excludable sets, which have sizes 19 and 20, respectively.

Given , a *k*-*proper partition* of a graph *G* is a partition of such that each part *P* of induces a *k*-connected subgraph of *G*. We prove that if *G* is a graph of order *n* such that , then *G* has a 2-proper partition with at most parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that if *G* is a graph of order *n* with minimum degree

where , then *G* has a *k*-proper partition into at most parts. This improves a result of Ferrara et al. ( Discrete Math 313 (2013), 760–764), and both the degree condition and the number of parts is best possible up to the constant *c*.

Suppose and are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on *M* and *N* for the existence of a simple graph *G*, which admits a face 2-colorable planar embedding in which the faces of one color have boundary lengths and the faces of the other color have boundary lengths . Such a graph is said to have a planar -biembedding. We also determine necessary and sufficient conditions on *M* and *N* for the existence of a simple graph *G* whose edge set can be partitioned into *r* cycles of lengths and also into *t* cycles of lengths . Such a graph is said to be -decomposable.

Let *G* be a graph of minimum degree at least 2 with no induced subgraph isomorphic to *K*_{1, 6}. We prove that if *G* is not isomorphic to one of eight exceptional graphs, then it is possible to assign two-element subsets of to the vertices of *G* in such a way that for every and every vertex the label *i* is assigned to *v* or one of its neighbors. It follows that *G* has fractional domatic number at least 5/2. This is motivated by a problem in robotics and generalizes a result of Fujita, Yamashita, and Kameda who proved that the same conclusion holds for all 3-regular graphs.

There are numerous results bounding the circumference of certain 3-connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3-connected graph, however. Oporowski, Oxley, and Thomas (J Combin Theory Ser B 57 (1993), 2, 239–257) proved the following result in 1993. For every positive integer *k*, there is an integer such that every 3-connected graph with at least *n* vertices contains a - or -minor. This result implies that the size of the largest bond in a 3-connected graph grows with the order of the graph. Oporowski et al. obtained a huge function iteratively. In this article, we first improve the above authors' result and provide a significantly smaller and simpler function . We then use the result to obtain a lower bound for the largest bond of a 3-connected graph by showing that any 3-connected graph on *n* vertices has a bond of size at least . In addition, we show the following: Let *G* be a 3-connected planar or cubic graph on *n* vertices. Then for any , *G* has a -minor with , and thus a bond of size at least .

We prove that every digraph of circumference *l* has DAG-width at most *l*. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).^{1} As a consequence of this result we deduce that the *k*-linkage problem is polynomially solvable for every fixed *k* in the class of digraphs with bounded circumference. This answers a question posed in J. Bang-Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak *k*-linkage problem (where we ask for arc-disjoint paths) is polynomially solvable for every fixed *k* in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP-hard on digraphs of DAG-width at most 5.

A spanning subgraph *F* of a graph *G* is called *perfect* if *F* is a forest, the degree of each vertex *x* in *F* is odd, and each tree of *F* is an induced subgraph of *G*. We provide a short linear-algebraic proof of the following theorem of A. D. Scott (Graphs Combin 17 (2001), 539–553): A connected graph *G* contains a perfect forest if and only if *G* has an even number of vertices.

We characterize the quartic (i.e., 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from these by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every edge in a triangle is either the square of a cycle, the line graph of a cubic graph or a graph obtained from the line multigraph of a cubic multigraph by replacing triangles with copies of *K*_{1, 1, 3}.

The clique number of an undirected graph *G* is the maximum order of a complete subgraph of *G* and is a well-known lower bound for the chromatic number of *G*. Every proper *k*-coloring of *G* may be viewed as a homomorphism (an edge-preserving vertex mapping) of *G* to the complete graph of order *k*. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth *k* for all .

Let *H* denote the tree with six vertices, two of which are adjacent and of degree 3. Let *G* be a graph and be distinct vertices of *G*. We characterize those *G* that contain a topological *H* in which are of degree 3 and are of degree 1, which include all 5-connected graphs. This work was motivated by the Kelmans–Seymour conjecture that 5-connected nonplanar graphs contain topological *K*_{5}.

Determining the maximum number of edges in an *n*-vertex *C*_{4}-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of *C*_{4}-free graphs are the Erdős-Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Rényi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Rényi orthogonal polarity graph.

We seek the maximum number of colors in an edge-coloring of the complete graph not having *t* edge-disjoint rainbow spanning subgraphs of specified types. Let , , and denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove for and for . We prove for and for . We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than *q* edges at a vertex.

Let *B* be a positive integer and let *G* be a simple graph. An excessive [*B*]-factorization of *G* is a minimum set of matchings, each of size *B*, whose union is . The number of matchings in an excessive [*B*]-factorization of *G* (or ∞ if an excessive [*B*]-factorization does not exist) is a graph parameter called the *excessive* [*B*]-*index* of *G* and denoted by . In this article we prove that, for any fixed value of *B*, the parameter can be computed in polynomial time in the size of the graph *G*. This solves a problem posed by one of the authors at the 21st British Combinatorial Conference.

Given a graph *G* of order *n*, the σ-*polynomial* of *G* is the generating function where is the number of partitions of the vertex set of *G* into *i* nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ-polynomials of graphs with chromatic number at least had all real roots, and conjectured the same held for chromatic number . We affirm this conjecture.

In Aldred and Plummer (Discrete Math 197/198 (1999) 29–40) proved that every *m*-connected -free graph of even order has a perfect matching *M* with and , where *F*_{1} and *F*_{2} are prescribed disjoint sets of independent edges with and . It is known that if *l* satisfies , then the star-free condition in the above result is best possible. In this paper, for , we prove a refinement of the result in which the condition is replaced by the weaker condition that *G* is -free (note that the new condition does not depend on *l*). We also show that if *m* is even and either or , then for *m*-connected graphs *G* with sufficiently large order, one can replace the condition by the still weaker condition that *G* is -free. The star-free conditions in our results are best possible.

A graph *G* is -*colorable* if can be partitioned into two sets and so that the maximum degree of is at most *j* and of is at most *k*. While the problem of verifying whether a graph is (0, 0)-colorable is easy, the similar problem with in place of (0, 0) is NP-complete for all nonnegative *j* and *k* with . Let denote the supremum of all *x* such that for some constant every graph *G* with girth *g* and for every is -colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing *g* from 5 further on does not increase much. Our constructions show that for every *g*, . We also find exact values of for all *g* and all .

A *bi-subdivision* of a graph *J* is a graph *H* obtained from *J* by subdividing each of its edges by inserting an even number of vertices. A matching covered subgraph *H* of a matching covered graph *G* is *conformal* if has a perfect matching. Using the theory of ear decompositions, Lovász (Combinatorica, 3 (1983), 105–117) showed that every nonbipartite matching covered graph has a conformal subgraph which is either a bi-subdivision of *K*_{4} or of . (The graph is the triangular prism.) A matching covered graph is *K*_{4}-*based* if it contains a bi-subdivision of *K*_{4} as a conformal subgraph; otherwise it is *K*_{4}-*free*. -*based* and -*free*
graphs are analogously defined. The result of Lovász quoted above implies that any nonbipartite matching covered graph is either *K*_{4}-based or -based (or both). The problem of deciding which matching covered graphs are *K*_{4}-based and which are -based is, in general, unsolved. In this paper, we present a solution to this classification problem in the special case of planar graphs. In Section 2, we show that a matching covered graph is *K*_{4}-free (-free) if and only if each of its bricks is *K*_{4}-free (-free). In Section 5, we show that a planar brick is *K*_{4}-free if and only if it has precisely two odd faces. In Section 6, we determine the list of all -free planar bricks; apart from one exception, it consists of two infinite families of bricks. The principal tool we use for proving our results is the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769–817).

Let *D* be a digraph and let be the arc-strong connectivity of *D*, and be the size of a maximum matching of *D*. We proved that if , then *D* has a spanning eulerian subdigraph.

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is *G*-symmetric if *G* is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by *G*, then Γ is an imprimitive *G*-symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.

We show that every *n*-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any -vertex planar graph. In order to achieve this result, we prove that every *n*-vertex plane graph has an induced outerplane subgraph containing at least vertices. Also, we show that every *n*-vertex planar graph and every *n*-vertex planar partial 3-tree admit a simultaneous embedding without mapping and with fixed edges.

Consider a graph *G* on *n* vertices satisfying the following Ore-type condition: for any two nonadjacent vertices *x* and *y* of *G*, we have . We conjecture that if we color the edges of *G* with two colors then the vertex set of *G* can be partitioned to two vertex disjoint monochromatic cycles of distinct colors. In this article, we prove an asymptotic version of this conjecture.

We produce an edge-coloring of the complete 3-uniform hypergraph on *n* vertices with colors such that the edges spanned by every set of five vertices receive at least three distinct colors. This answers the first open case of a question of Conlon-Fox-Lee-Sudakov (Int Math Res Not, to appear) who asked whether such a coloring exists with colors.

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck—the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph *G* and any finite sequence of graphs, it gives a linear constraint that every reconstruction of *G* must satisfy. Let be the number of distinct (mutually nonisomorphic) graphs on *n* vertices, and let be the number of distinct decks that can be constructed from these graphs. Then the difference measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for *n*-vertex graphs if and only if . We give a framework based on Kocay's lemma to study this discrepancy. We prove that if *M* is a matrix of covering numbers of graphs by sequences of graphs, then . In particular, all *n*-vertex graphs are reconstructible if one such matrix has rank . To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix *M* of covering numbers satisfies .

A graph *G* is *perfect* if for all induced subgraphs *H* of *G*, . A graph *G* is *Berge* if neither *G* nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew-partition.” A *clique-coloring* of a graph *G* is an assignment of colors to the vertices of *G* in such a way that no inclusion-wise maximal clique of *G* of size at least two is monochromatic, and the *clique-chromatic number* of *G*, denoted by , is the smallest number of colors needed to clique-color *G*. There exist graphs of arbitrarily large clique-chromatic number, but it is not known whether the clique-chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew-partition is 2-clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. (http://arxiv.org/abs/1308.6444).

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.

Given a graph and a colouring , the induced colour of a vertex *v* is the sum of the colours at the edges incident with *v*. If all the induced colours of vertices of *G* are distinct, the colouring is called antimagic. If *G* has a bijective antimagic colouring , the graph *G* is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than *K*_{2} are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least for some constant *c*; we improve on this result, proving the conjecture for graphs with average degree at least some constant *d*_{0}.

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 − Laplacian Δ_{1}. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ_{1} eigenvalue for connected graphs.

An edge coloring of a graph is said to be an *r*-local coloring if the edges incident to any vertex are colored with at most *r* colors. Generalizing a result of Bessy and Thomassé, we prove that the vertex set of any 2-locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number *r*, we show that the vertex set of any *r*-locally colored complete graph may be partitioned into disjoint monochromatic cycles. This generalizes a result of Erdős, Gyárfás, and Pyber.

The global mean of subtrees of a tree is the average order (i.e., average number of vertices) of its subtrees. Analogously, the local mean of a vertex in a tree is the average order of subtrees containing this vertex. In the comprehensive study of these concepts by Jamison (J Combin Theory Ser B 35 (1983), 207–223 and J Combin Theory Ser B 37 (1984), 70–78), several open questions were proposed. One of them asks if the largest local mean always occurs at a leaf vertex. Another asks if it is true that the local mean of any vertex of any tree is at most twice the global mean. In this note, we answer the first question by showing that the largest local mean always occurs at a leaf or a vertex of degree 2 and that both cases are possible. With this result, a positive answer to the second question is provided. We also show some related results on local mean and global mean of trees.

Let *G* be a graph with vertex set and let be a set function associated with *G*. An *H*-factor of graph *G* is a spanning subgraphs *F* such that

Let be an even integer-valued function such that and let for . In this article, we investigate -factors of graphs by using Lovász's structural descriptions. Let denote the number of odd components of *G*. We show that if one of the following conditions holds, then *G* contains an -factor.

- (i)is even and for all ;
- (ii)is odd, for all and for all .

As a corollary, we show that if a graph *G* of odd order with minimum degree at least satisfies

then *G* contains an -factor, where . In particular, we make progress on the characterization problem for a special family of graphs proposed by Akiyama and Kano.

Let and denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph *G*, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph *G* and bounds of , Cioabă and Wong conjectured that for any integers and a *d*-regular graph *G*, if , then . They proved the conjecture for , and presented evidence for the cases when . Thus the conjecture remains open for . We propose a more general conjecture that for a graph *G* with minimum degree , if , then . In this article, we prove that for a graph *G* with minimum degree δ, each of the following holds.

- (i)For , if and , then .
- (ii)For , if and , then .

Our results sharpen theorems of Cioabă and Wong and give a partial solution to Cioabă and Wong's conjecture and Seymour's problem. We also prove that for a graph *G* with minimum degree , if , then the edge connectivity is at least *k*, which generalizes a former result of Cioabă. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on and edge connectivity.

In 1998 the second author proved that there is an such that every graph satisfies . The first author recently proved that any graph satisfying contains a stable set intersecting every maximum clique. In this note, we exploit the latter result to give a much shorter, simpler proof of the former. Working from first principles, we omit only some five pages of proofs of known intermediate results (which appear in an extended version of this paper), and the proofs of Hall's Theorem, Brooks' Theorem, the Lovász Local Lemma, and Talagrand's Inequality.

A graph *G* is equimatchable if each matching in *G* is a subset of a maximum-size matching and it is factor critical if has a perfect matching for each vertex *v* of *G*. It is known that any 2-connected equimatchable graph is either bipartite or factor critical. We prove that for 2-connected factor-critical equimatchable graph *G* the graph is either or for some *n* for any vertex *v* of *G* and any minimal matching *M* such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus *g* to if and to if . Moreover, for any nonnegative integer *g* we construct a 2-connected equimatchable factor-critical graph with genus *g* and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers *g*, *h*, and *k* we provide a construction of arbitrarily large 2-connected equimatchable bipartite graphs with orientable genus *g*, respectively nonorientable genus *h*, and a genus embedding with face-width *k*. Finally, we prove that any *d*-degenerate 2-connected equimatchable factor-critical graph has at most vertices, where a graph is *d*-degenerate if every its induced subgraph contains a vertex of degree at most *d*.

Biregular -cages are graphs of girth *g* that contain vertices of degrees *r* and *m* and are of the smallest order among all such graphs. We show that for every and every odd , there exists an integer *m*_{0} such that for every *even* , the biregular -cage is of order equal to a natural lower bound analogous to the well-known Moore bound. In addition, when *r* is odd, the restriction on the parity of *m* can be removed, and there exists an integer *m*_{0} such that a biregular -cage of order equal to this lower bound exists *for all* . This is in stark contrast to the result classifying all cages of degree *k* and girth *g* whose order is equal to the Moore bound.

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.

A star edge coloring of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this article, we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.

A perfect matching covering of a graph *G* is a set of perfect matchings of *G* such that every edge of *G* is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of *G*). A cubic graph *G* is called a Kotzig graph if *G* has a 3-edge-coloring such that each pair of colors forms a hamiltonian circuit (introduced by R. Häggkvist, K. Markström, J Combin Theory Ser B 96 (2006), 183–206). In this article, we prove that if there is a vertex *w* of a cubic graph *G* such that , the graph obtained from by suppressing all degree two vertices is a Kotzig graph, then *G* has a Berge covering. We also obtain some results concerning the so-called 5-even subgraph double cover conjecture.

A *clique covering* of a simple graph *G* is a collection of cliques of *G* covering all the edges of *G* such that each vertex is contained in at most *k* cliques. The smallest *k* for which *G* admits a clique covering is called the local clique cover number of *G* and is denoted by lcc(*G*). Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well-known problems are discussed. In particular, it is proved that the local clique cover number of every claw-free graph is at most , where Δ is the maximum degree of the graph and *c* is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen et al. (Clique covering the edges of a locally cobipartite graph, Discrete Math 219(1–3)(2000), 17–26), we prove that the clique cover number of every connected claw-free graph on *n* vertices with the minimum degree δ, is at most , where *c* is a constant.