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

Two Hamilton paths in are separated by a cycle of length *k* if their union contains such a cycle. For we bound the asymptotics of the maximum cardinality of a family of Hamilton paths in such that any pair of paths in the family is separated by a cycle of length *k*. We also deal with related problems, including directed Hamilton paths.

The choosability of a graph *G* is the minimum *k* such that having *k* colors available at each vertex guarantees a proper coloring. Given a toroidal graph *G*, it is known that , and if and only if *G* contains *K*_{7}. Cai et al. (J Graph Theory 65(1) (2010), 1–15) proved that a toroidal graph *G* without 7-cycles is 6-choosable, and if and only if *G* contains *K*_{6}. They also proved that a toroidal graph *G* without 6-cycles is 5-choosable, and conjectured that if and only if *G* contains *K*_{5}. We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither *K*_{5} nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither (a *K*_{5} missing one edge) nor 6-cycles are 4-choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4-choosable.

For any graph *G*, let be the number of spanning trees of *G*, be the line graph of *G*, and for any nonnegative integer *r*, be the graph obtained from *G* by replacing each edge *e* by a path of length connecting the two ends of *e*. In this article, we obtain an expression for in terms of spanning trees of *G* by a combinatorial approach. This result generalizes some known results on the relation between and and gives an explicit expression if *G* is of order and size in which *s* vertices are of degree 1 and the others are of degree *k*. Thus we prove a conjecture on for such a graph *G*.

Given a graph *F*, a graph *G* is *uniquely F*-*saturated* if *F* is not a subgraph of *G* and adding any edge of the complement to *G* completes exactly one copy of *F*. In this article, we study uniquely -saturated graphs. We prove the following: (1) a graph is uniquely *C*_{5}-saturated if and only if it is a friendship graph. (2) There are no uniquely *C*_{6}-saturated graphs or uniquely *C*_{7}-saturated graphs. (3) For , there are only finitely many uniquely -saturated graphs (we conjecture that in fact there are none). Additionally, our results show that there are finitely many *k*-friendship graphs (as defined by Kotzig) for .

We prove that for every fixed *k*, the number of occurrences of the transitive tournament Tr_{k} of order *k* in a tournament on *n* vertices is asymptotically minimized when is random. In the opposite direction, we show that any sequence of tournaments achieving this minimum for any fixed is necessarily quasirandom. We present several other characterizations of quasirandom tournaments nicely complementing previously known results and relatively easily following from our proof techniques.

We construct for all a *k*-edge-connected digraph *D* with such that there are no edge-disjoint and paths. We use in our construction “self-similar” graphs which technique could be useful in other problems as well.

In this article we prove a new result about partitioning colored complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for , in every edge coloring of with the colors red and blue, it is possible to cover all the vertices with *k* disjoint red paths and a disjoint blue balanced complete -partite graph. When the coloring of is connected in red, we prove a stronger result—that it is possible to cover all the vertices with *k* red paths and a blue balanced complete -partite graph. Using these results we determine the Ramsey number of an *n*-vertex path, , versus a balanced complete *t*-partite graph on vertices, , whenever . We show that in this case , generalizing a result of Erdős who proved the case of this result. We also determine the Ramsey number of a path versus the power of a path . We show that , solving a conjecture of Allen, Brightwell, and Skokan.

A drawing of a graph is *pseudolinear* if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The *pseudolinear crossing number* of a graph *G* is the minimum number of pairwise crossings of edges in a pseudolinear drawing of *G*. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.

A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity 5, and thus girth 5. We show by computer search that the smallest members of this class are three graphs with 76 vertices.

We study the existence and the number of *k*-dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erdős and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of *k*-dominating independent sets in *n*-vertex graphs is between and if , moreover the maximum number of 2-dominating independent sets in *n*-vertex graphs is between and . Graph constructions containing a large number of *k*-dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.

A natural topic of algebraic graph theory is the study of vertex transitive graphs. In the present article, we investigate locally 3-transitive graphs of girth 4. Taking our former results on locally symmetric graphs of girth 4 as a starting point, we show what properties are retained if we weaken the requirement of local symmetry to local 3-transitivity.

Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This article surveys several graph parameters tied to treewidth, including separation number, tangle number, well-linked number, and Cartesian tree product number. We review many results in the literature showing these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.

Given graphs *H* and *F*, a subgraph is an *F*-*saturated subgraph* of *H* if , but for all . The *saturation number of F in H*, denoted , is the minimum number of edges in an *F*-saturated subgraph of *H*. In this article, we study saturation numbers of tripartite graphs in tripartite graphs. For and *n*_{1}, *n*_{2}, and *n*_{3} sufficiently large, we determine and exactly and within an additive constant. We also include general constructions of -saturated subgraphs of with few edges for .

The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique-width if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permutation classes.

The crossing number cr(*G*) of a graph *G* is the minimum number of crossings in a drawing of *G* in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number of *G* is the minimum number of crossings in a such drawing of *G* with edges as straight line segments. Zarankiewicz proved in 1952 that . We generalize the upper bound to

and prove . We also show that for *n* large enough, and , with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete *r*-partite graph. Richter and Thomassen proved in 1997 that the limit as of over the maximum number of crossings in a drawing of exists and is at most . We define and show that for a fixed *r* and the balanced complete *r*-partite graph, is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

*Equistable graphs* are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 1. *Strongly equistable graphs* are graphs such that for every and every nonempty subset *T* of vertices that is not a maximal stable set, there exist positive vertex weights assigning weight 1 to every maximal stable set such that the total weight of *T* does not equal *c*. *General partition graphs* are the intersection graphs of set systems over a finite ground set *U* such that every maximal stable set of the graph corresponds to a partition of *U*. General partition graphs are exactly the graphs every edge of which is contained in a strong clique. In 1994, Mahadev, Peled, and Sun proved that every strongly equistable graph is equistable, and conjectured that the converse holds as well. In 2009, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin's conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An “intermediate” conjecture, posed by Miklavič and Milanič in 2011, states that every equistable graph has a strong clique. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle-free graphs. We also show that not all strongly equistable graphs are general partition.

A graph is -colorable if its vertex set can be partitioned into *r* sets so that the maximum degree of the graph induced by is at most for each . For a given pair , the question of determining the minimum such that planar graphs with girth at least *g* are -colorable has attracted much interest. The finiteness of was known for all cases except when . Montassier and Ochem explicitly asked if *d*_{2}(5, 1) is finite. We answer this question in the affirmative with ; namely, we prove that all planar graphs with girth at least five are (1, 10)-colorable. Moreover, our proof extends to the statement that for any surface *S* of Euler genus γ, there exists a where graphs with girth at least five that are embeddable on *S* are (1, *K*)-colorable. On the other hand, there is no finite *k* where planar graphs (and thus embeddable on any surface) with girth at least five are (0, *k*)-colorable.

We show that a *k*-edge-connected graph on *n* vertices has at least spanning trees. This bound is tight if *k* is even and the extremal graph is the *n*-cycle with edge multiplicities . For *k* odd, however, there is a lower bound , where . Specifically, and . Not surprisingly, *c*_{3} is smaller than the corresponding number for 4-edge-connected graphs. Examples show that . However, we have no examples of 5-edge-connected graphs with fewer spanning trees than the *n*-cycle with all edge multiplicities (except one) equal to 3, which is almost 6-regular. We have no examples of 5-regular 5-edge-connected graphs with fewer than spanning trees, which is more than the corresponding number for 6-regular 6-edge-connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.

We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.

For a positive integer *k*, a *k*-*coloring* of a graph is a mapping such that whenever . The COLORING problem is to decide, for a given *G* and *k*, whether a *k*-coloring of *G* exists. If *k* is fixed (i.e., it is not part of the input), we have the decision problem *k*-COLORING instead. We survey known results on the computational complexity of COLORING and *k*-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.

Neumann-Lara (1985) and Škrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees, we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are “easier to reach” by a random walk, but “more difficult to get out of.” We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behavior or random walk on a graph to its eigenvalues in a new perspective.

A *dominating path* in a graph is a path *P* such that every vertex outside *P* has a neighbor on *P*. A result of Broersma from 1988 implies that if *G* is an *n*-vertex *k*-connected graph and , then *G* contains a dominating path. We prove the following results. The lengths of dominating paths include all values from the shortest up to at least . For , where *a* is a constant greater than 1/3, the minimum length of a dominating path is at most logarithmic in *n* when *n* is sufficiently large (the base of the logarithm depends on *a*). The preceding results are sharp. For constant *s* and , an *s*-vertex dominating path is guaranteed by when *n* is sufficiently large, but (where ) does not even guarantee a dominating set of size *s*. We also obtain minimum-degree conditions for the existence of a spanning tree obtained from a dominating path by giving the same number of leaf neighbors to each vertex.

The *minimum leaf number* ml(*G*) of a connected graph *G* is defined as the minimum number of leaves of the spanning trees of *G* if *G* is not hamiltonian and 1 if *G* is hamiltonian. We study nonhamiltonian graphs with the property for each or for each . These graphs will be called -leaf-critical *and l*-leaf-stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3-leaf-critical graphs (that turn out to be the so-called hypotraceable graphs) was an open problem until 1975. We show that *l*-leaf-stable and *l*-leaf-critical graphs exist for every integer , moreover for *n* sufficiently large, planar *l*-leaf-stable and *l*-leaf-critical graphs exist on *n* vertices. We also characterize 2-fragments of leaf-critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf-critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.

In this article, we define and study a new family of graphs that generalizes the notions of line graphs and path graphs. Let *G* be a graph with no loops but possibly with parallel edges. An ℓ*-link* of *G* is a walk of *G* of length in which consecutive edges are different. The ℓ*-link graph* of *G* is the graph with vertices the ℓ-links of *G*, such that two vertices are joined by edges in if they correspond to two subsequences of each of μ -links of *G*. By revealing a recursive structure, we bound from above the chromatic number of ℓ-link graphs. As a corollary, for a given graph *G* and large enough ℓ, is 3-colorable. By investigating the shunting of ℓ-links in *G*, we show that the Hadwiger number of a nonempty is greater or equal to that of *G*. Hadwiger's conjecture states that the Hadwiger number of a graph is at least the chromatic number of that graph. The conjecture has been proved by Reed and Seymour (Eur J Combin 25(6) (2004), 873–876) for line graphs, and hence 1-link graphs. We prove the conjecture for a wide class of ℓ-link graphs.

For a graph , let denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of *G* so that each edge of *G* belongs to exactly one of them. It is easy to see that for every graph *G*, , where is the maximum size of an independent set of *G*. Erdős conjectured in the 80s that for almost every graph *G* equality holds, that is that for the random graph , with high probability, that is with probability that tends to 1 as *n* tends to infinity. The first author showed that this is slightly false, proving that for most values of *n* tending to infinity and for , with high probability. We prove a stronger bound: there exists an absolute constant so that with high probability.

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices.

In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs *G* with *n* vertices and , which has the most complete subgraphs of size *t*, for . The conjectured extremal graph is , where with . Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when , and also reduced the general conjecture to the case . We prove the conjecture for and also establish a weaker form of the conjecture for all *r*.

We show that the 4-coloring problem can be solved in polynomial time for graphs with no induced 5-cycle *C*_{5} and no induced 6-vertex path *P*_{6}

We prove that a graph *G* contains no induced -vertex path and no induced complement of a -vertex path if and only if *G* is obtained from 5-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class-preserving operation introduced here.

Let be a sequence of of nonnegative integers pairs. If a digraph *D* with satisfies and for each *i* with , then **d** is called a degree sequence of *D*. If *D* is a strict digraph, then **d** is called a strict digraphic sequence. Let be the collection of digraphs with degree sequence **d**. We characterize strict digraphic sequences **d**
for which there exists a strict strong digraph .

Let *T* be a tournament of order *n* and be the number of cycles of length *m* in *T*. For and odd *n*, the maximum of is achieved for any regular tournament of order *n* (M. G. Kendall and B. Babington Smith, ), and in the case it is attained only for the unique regular locally transitive tournament *RLT*_{n} of order *n* (U. Colombo, ). A lower bound was also obtained for in the class of regular tournaments of order *n* (A. Kotzig, ). This bound is attained if and only if *T* is doubly regular (when ) or nearly doubly regular (when ) (B. Alspach and C. Tabib, ). In the present article, we show that for any regular tournament *T* of order *n*, the equality holds. This allows us to reduce the case to the case In turn, the pure spectral expression for obtained in the class implies that for a regular tournament *T* of order the inequality holds, with equality if and only if *T* is doubly regular or *T* is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of *c*_{6}(*RLT*_{n}) and conjecture that this value coincides with the minimum number of 6-cycles in the class for each odd

We prove that the number of 1-factorizations of a generalized Petersen graph of the type is equal to the *k*th Jacobsthal number when *k* is odd, and equal to when *k* is even. Moreover, we verify the list coloring conjecture for .

Let *H* be a given graph. A graph *G* is said to be *H*-free if *G* contains no induced copies of *H*. For a class of graphs, the graph *G* is -free if *G* is *H*-free for every . Bedrossian characterized all the pairs of connected subgraphs such that every 2-connected -free graph is hamiltonian. Faudree and Gould extended Bedrossian's result by proving the necessity part of the result based on infinite families of non-hamiltonian graphs. In this article, we characterize all pairs of (not necessarily connected) graphs such that there exists an integer *n*_{0} such that every 2-connected -free graph of order at least *n*_{0} is hamiltonian.

A *k*-hypertournament *H* on *n* vertices () is a pair , where *V* is the vertex set of *H* and *A* is a set of *k*-tuples of vertices, called arcs, such that for all subsets with , *A* contains exactly one permutation of *S* as an arc. Recently, Li et al. showed that any strong *k*-hypertournament *H* on *n* vertices, where , is vertex-pancyclic, an extension of Moon's theorem for tournaments. In this article, we examine several generalizations of regular tournaments and prove the following generalization of Alspach's theorem concerning arc-pancyclicity: Every Σ-regular *k*-hypertournament on *n* vertices, where , is arc-pancyclic.

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least five in this class is 3-colorable.

A graph is *strongly even-cycle decomposable* if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.

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*. We study lists by three 1-factors, and call with a -core of *G*. If *G* is not 3-edge-colorable, then . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that if , then is an upper bound for the girth of *G*. We show that bounds the oddness of *G* as well. We prove that . If , then every -core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4-edge-connected cubic graph *G* with . On the other hand, the difference between and can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer , there exists a bridgeless cubic graph *G* such that .

A class of graphs is *hereditary* if it is closed under isomorphism and induced subgraphs. A class of graphs is χ*-bounded* if there exists a function such that for all graphs , and all induced subgraphs *H* of *G*, we have that . We prove that proper homogeneous sets, clique-cutsets, and amalgams together preserve χ-boundedness. More precisely, we show that if and are hereditary classes of graphs such that is χ-bounded, and such that every graph in either belongs to or admits a proper homogeneous set, a clique-cutset, or an amalgam, then the class is χ-bounded. This generalizes a result of [*J Combin Theory Ser B* 103(5) (), 567–586], which states that proper homogeneous sets and clique-cutsets together preserve χ-boundedness, as well as a result of [*European J Combin* 33(4) (), 679–683], which states that 1-joins preserve χ-boundedness. The *house* is the complement of the four-edge path. As an application of our result and of the decomposition theorem for “cap-free” graphs from [*J Graph Theory* 30(4) (), 289–308], we obtain that if *G* is a graph that does not contain any subdivision of the house as an induced subgraph, then .

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible.

Král' and Sgall (J Graph Theory 49(3) (2005), 177–186) introduced a refinement of list coloring where every color list must be subset to one predetermined palette of colors. We call this -*choosability* when the palette is of size at most ℓ and the lists must be of size at least *k*. They showed that, for any integer , there is an integer , satisfying as , such that, if a graph is -choosable, then it is *C*-choosable, and asked if *C* is required to be exponential in *k*. We demonstrate it must satisfy . For an integer , if is the least integer such that a graph is -choosable if it is -choosable, then we more generally supply a lower bound on , one that is super-polynomial in *k* if , by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on that improve on earlier bounds if .

The **dicycle transversal number** of a digraph *D* is the minimum size of a **dicycle transversal** of *D*, that is a set of vertices of *D*, whose removal from *D* makes it acyclic.

An arc *a* of a digraph *D* with at least one cycle is a **transversal arc** if *a* is in every directed cycle of *D* (making acyclic). In [3] and [4], we completely characterized the complexity of following problem: Given a digraph *D*, decide if there is a dicycle *B* in *D* and a cycle *C* in its underlying undirected graph such that . It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). In the remaining case (allowing arbitrarily many transversal vertices) the problem is NP-complete. In this article, we classify the complexity of the arc-analog of this problem, where we ask for a dicycle *B* and a cycle *C* that are arc-disjoint, but not necessarily vertex-disjoint. We prove that the problem is polynomially solvable for strong digraphs and for digraphs with a constantly bounded number of transversal arcs (but possibly an unbounded number of transversal vertices). In the remaining case (allowing arbitrarily many transversal arcs) the problem is NP-complete.

Hadwiger's conjecture asserts that every graph with chromatic number *t* contains a complete minor of order *t*. Given integers , the Kneser graph is the graph with vertices the *k*-subsets of an *n*-set such that two vertices are adjacent if and only if the corresponding *k*-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.

Let be an integer and let *D* be a simple digraph on vertices. We prove that If

then *D* must have a nontrivial subdigraph *H* such that the strong arc connectivity of *H* is at least . We also show that this bound is best possible and present a constructive characterization for the extremal graphs.

The celebrated grid exclusion theorem states that for every *h*-vertex planar graph *H*, there is a constant such that if a graph *G* does not contain *H* as a minor then *G* has treewidth at most . We are looking for patterns of *H* where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel , the double wheel , any graph of pathwidth at most 2 , and the yurt graph .

We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Barát and Thomassen and is a step toward a generalization of Schnyder woods to higher genus surfaces.

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.

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.

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.

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.

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.

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.

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.

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.

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*.