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

Let *c* be a proper edge coloring of a graph with integers . Then , while Vizing's theorem guarantees that we can take . On the course of investigating irregularities in graphs, it has been conjectured that with only slightly larger *k*, that is, , we could enforce an additional strong feature of *c*, namely that it attributes distinct sums of incident colors to adjacent vertices in *G* if only this graph has no isolated edges and is not isomorphic to *C*_{5}. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact an even stronger statement holds, as the necessary number of colors stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph *G* of maximum degree at least 28, which contains no isolated edges admits a proper edge coloring such that for every edge of *G*.

This article introduces a new variant of hypercubes . The *n*-dimensional twisted hypercube is obtained from two copies of the -dimensional twisted hypercube by adding a perfect matching between the vertices of these two copies of . We prove that the *n*-dimensional twisted hypercube has diameter . This improves on the previous known variants of hypercube of dimension *n* and is optimal up to an error of order . Another type of hypercube variant that has similar structure and properties as is also discussed in the last section.

The *k*-linkage problem is as follows: given a digraph and a collection of *k* terminal pairs such that all these vertices are distinct; decide whether *D* has a collection of vertex disjoint paths such that is from to for . A digraph is *k*-linked if it has a *k*-linkage for every choice of 2*k* distinct vertices and every choice of *k* pairs as above. The *k*-linkage problem is NP-complete already for [11] and there exists no function such that every -strong digraph has a *k*-linkage for every choice of 2*k* distinct vertices of *D* [17]. Recently, Chudnovsky et al. [9] gave a polynomial algorithm for the *k*-linkage problem for any fixed *k* in (a generalization of) semicomplete multipartite digraphs. In this article, we use their result as well as the classical polynomial algorithm for the case of acyclic digraphs by Fortune et al. [11] to develop polynomial algorithms for the *k*-linkage problem in locally semicomplete digraphs and several classes of decomposable digraphs, including quasi-transitive digraphs and directed cographs. We also prove that the necessary condition of being -strong is also sufficient for round-decomposable digraphs to be *k*-linked, obtaining thus a best possible bound that improves a previous one of . Finally we settle a conjecture from [3] by proving that every 5-strong locally semicomplete digraph is 2-linked. This bound is also best possible (already for tournaments) [1].

Recently, Borodin, Kostochka, and Yancey (Discrete Math 313(22) (2013), 2638–2649) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in colors such that each monochromatic component is a path of length at most 14. Moreover, we show a list version of this result. On the other hand, for each positive integer , we construct a planar graph of girth 4 such that in any coloring of vertices in colors there is a monochromatic path of length at least *t*. It remains open whether each planar graph of girth 5 admits a 2-coloring with no long monochromatic paths.

A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, *K*_{7} and the 13 graphs obtained from *K*_{7} by moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1, 9, 11, 12]. This set includes exactly one bipartite graph, the Heawood graph. In this article we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the *K*_{3, 3, 1, 1} and families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.

Hedetniemi conjectured in 1966 that if *G* and *H* are finite graphs with chromatic number *n*, then the chromatic number of the direct product of *G* and *H* is also *n*. We mention two well-known results pertaining to this conjecture and offer an improvement of the one, which partially proves the other. The first of these two results is due to Burr et al. (Ars Combin 1 (1976), 167–190), who showed that when every vertex of a graph *G* with is contained in an *n*-clique, then whenever . The second, by Duffus et al. (J Graph Theory 9 (1985), 487–495), and, obtained independently by Welzl (J Combin Theory Ser B 37 (1984), 235–244), states that the same is true when *G* and *H* are connected graphs each with clique number *n*. Our main result reads as follows: If *G* is a graph with and has the property that the subgraph of *G* induced by those vertices of *G* that are not contained in an *n*-clique is homomorphic to an -critical graph *H*, then . This result is an improvement of the result by the first authors. In addition we will show that our main result implies a special case of the result by the second set of authors. Our approach will employ a construction of a graph *F*, with chromatic number , that is homomorphic to *G* and *H*.

Let and be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree *d* and diameter *k*. When , it is well known that with equality if and only if the graph is a Moore graph. In the abelian case, we have . The best currently lower bound on is for all sufficiently large *d*. In this article, we consider the construction of large graphs of diameter 2 using generalized difference sets. We show that for sufficiently large *d* and if , and *m* is odd.

Recently, Jones et al. (Electron J Comb 22(2) (2015), #P2.53) introduced the study of *u*-representable graphs, where *u* is a word over containing at least one 1. The notion of a *u*-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is 11⋅⋅⋅1-representable assuming that the number of 1s is at least three, while the class of 12-representable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda (M.Sc. thesis, 2015), who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of *u*-representable graphs at least for small values of *u*. In this article, we show that if *u* is of length at least 3 then any graph is *u*-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting nonequivalent cases in the theory of *u*-representable graphs, namely, those of and .

Let *F* be a graph that contains an edge whose deletion reduces its chromatic number. For such a graph *F*, a classical result of Simonovits from 1966 shows that every graph on vertices with more than edges contains a copy of *F*. In this article we derive a similar theorem for multipartite graphs. For a graph *H* and an integer , let be the minimum real number such that every ℓ-partite graph whose edge density between any two parts is greater than contains a copy of *H*. Our main contribution in this article is to show that for all sufficiently large if and only if *H* admits a vertex-coloring with colors such that all color classes but one are independent sets, and the exceptional class induces just a matching. When *H* is a complete graph, this recovers a result of Pfender (Combinatorica 32 (2012), 483–495). We also consider several extensions of Pfender's result.

Given a family and a host graph *H*, a graph is -*saturated relative to H* if no subgraph of *G* lies in but adding any edge from to *G* creates such a subgraph. In the -*saturation game* on *H*, players *Max* and *Min* alternately add edges of *H* to *G*, avoiding subgraphs in , until *G* becomes -saturated relative to *H*. They aim to maximize or minimize the length of the game, respectively; denotes the length under optimal play (when Max starts).

Let denote the family of odd cycles and the family of *n*-vertex trees, and write *F* for when . Our results include , for , for , and for . We also determine ; with , it is *n* when *n* is even, *m* when *n* is odd and *m* is even, and when is odd. Finally, we prove the lower bound . The results are very similar when Min plays first, except for the *P*_{4}-saturation game on .

Let *G* be a regular bipartite graph and . We show that there exist perfect matchings of *G* containing both, an odd and an even number of edges from *X* if and only if the signed graph , that is a graph *G* with exactly the edges from *X* being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2-cycle-cover such that each cycle contains an odd number of negative edges.

Given a digraph *G*, we propose a new method to find the recurrence equation for the number of vertices of the *k*-iterated line digraph , for , where . We obtain this result by using the minimal polynomial of a quotient digraph of *G*.

We consider random-turn positional games, introduced by Peres, Schramm, Sheffield, and Wilson in 2007. A *p*-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is *p*). We analyze the random-turn version of several classical Maker–Breaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the *k*-vertex-connectivity game (both played on the edge set of ). For each of these games we provide each of the players with a (randomized) efficient strategy that typically ensures his win in the asymptotic order of the minimum value of *p* for which he typically wins the game, assuming optimal strategies of both players.

A graph with a trivial automorphism group is said to be *rigid*. Wright proved (Acta Math 126(1) (1971), 1–9) that for a random graph is rigid whp (with high probability). It is not hard to see that this lower bound is sharp and for with positive probability is nontrivial. We show that in the sparser case , it holds whp that *G*'s 2-core is rigid. We conclude that for all *p*, a graph in is reconstructible whp. In addition this yields for a canonical labeling algorithm that almost surely runs in polynomial time with *o*(1) error rate. This extends the range for which such an algorithm is currently known (T. Czajka and G. Pandurangan, J Discrete Algorithms 6(1) (2008), 85–92).

We look at several saturation problems in complete balanced blow-ups of graphs. We let denote the blow-up of *H* onto parts of size *n* and refer to a copy of *H* in as *partite* if it has one vertex in each part of . We then ask how few edges a subgraph *G* of can have such that *G* has no partite copy of *H* but such that the addition of any new edge from creates a partite *H*. When *H* is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger in . Our main result is to calculate this value for when *n* is large. We also give exact results for paths and stars and show that for 2-connected graphs the answer is linear in *n* whilst for graphs that are not 2-connected the answer is quadratic in *n*. We also investigate a similar problem where *G* is permitted to contain partite copies of *H* but we require that the addition of any new edge from creates an extra partite copy of *H*. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.

A graph *G* is called -choosable if for any list assignment *L* that assigns to each vertex *v* a set of *a* permissible colors, there is a *b*-tuple *L*-coloring of *G*. An (*a*, 1)-choosable graph is also called *a*-choosable. In the pioneering article on list coloring of graphs by Erdős et al. , 2-choosable graphs are characterized. Confirming a special case of a conjecture in , Tuza and Voigt proved that 2-choosable graphs are -choosable for any positive integer *m*. On the other hand, Voigt proved that if *m* is an odd integer, then these are the only -choosable graphs; however, when *m* is even, there are -choosable graphs that are not 2-choosable. A graph is called 3-choosable-critical if it is not 2-choosable, but all its proper subgraphs are 2-choosable. Voigt conjectured that for every positive integer *m*, all bipartite 3-choosable-critical graphs are -choosable. In this article, we determine which 3-choosable-critical graphs are (4, 2)-choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer *k* such that for any positive integer *m*, every bipartite 3-choosable-critical graph is -choosable. Moving beyond 3-choosable-critical graphs, we present an infinite family of non-3-choosable-critical graphs that have been shown by computer analysis to be (4, 2)-choosable. This shows that the family of all (4, 2)-choosable graphs has rich structure.

Given two graphs *G* and *H*, an *H*-*decomposition* of *G* is a partition of the edge set of *G* such that each part is either a single edge or forms a graph isomorphic to *H*. Let be the smallest number ϕ such that any graph *G* of order *n* admits an *H*-decomposition with at most ϕ parts. Pikhurko and Sousa conjectured that for and all sufficiently large *n*, where denotes the maximum number of edges in a graph on *n* vertices not containing *H* as a subgraph. Their conjecture has been verified by Özkahya and Person for all edge-critical graphs *H*. In this article, the conjecture is verified for the *k*-fan graph. The *k*-*fan graph*, denoted by , is the graph on vertices consisting of *k* triangles that intersect in exactly one common vertex called the *center* of the *k*-fan.

For graphs *F* and *H*, we say *F* is *Ramsey for H* if every 2-coloring of the edges of *F* contains a monochromatic copy of *H*. The graph *F* is *Ramsey H*-*minimal* if *F* is Ramsey for *H* and there is no proper subgraph of *F* so that is Ramsey for *H*. Burr et al. defined to be the minimum degree of *F* over all Ramsey *H*-minimal graphs *F*. Define to be a graph on vertices consisting of a complete graph on *t* vertices and one additional vertex of degree *d*. We show that for all values ; it was previously known that , so it is surprising that is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that for all graphs *H*, where is the minimum degree of *H*; Szabó et al. investigated which graphs have this property and conjectured that all bipartite graphs *H* without isolated vertices satisfy . Fox et al. further conjectured that all connected triangle-free graphs with at least two vertices satisfy this property. We show that *d*-regular 3-connected triangle-free graphs *H*, with one extra technical constraint, satisfy ; the extra constraint is that *H* has a vertex *v* so that if one removes *v* and its neighborhood from *H*, the remainder is connected.

We give a complete description of the set of triples of real numbers with the following property. There exists a constant *K* such that is a lower bound for the matching number of every connected subcubic graph *G*, where denotes the number of vertices of degree *i* for each *i*.

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note, we show that for all integers , there exist oriented planar graphs of order *n* and digirth *g* for which the size of the maximum acyclic set is at most . When this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.

Tutte's 5-flow conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. It suffices to prove the conjecture for cyclically 6-edge-connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow. This implies that every minimum counterexample to the 5-flow conjecture has oddness at least 6.

In the *graph sharing game*, two players share a connected graph *G* with nonnegative weights assigned to the vertices claiming and collecting the vertices of *G* one by one, while keeping the set of all claimed vertices connected through the whole game. Each player wants to maximize the total weight of the vertices they have gathered by the end of the game, when the whole *G* has been claimed. It is proved that for any class of graphs with an odd number of vertices and with forbidden subdivision of a fixed graph (e.g., for the class of planar graphs with an odd number of vertices), there is a constant such that the first player can secure at least the proportion of the total weight of *G* whenever . Known examples show that such a constant does no longer exist if any of the two conditions on the class (an odd number of vertices or a forbidden subdivision) is removed. The main ingredient in the proof is a new structural result on weighted graphs with a forbidden subdivision.

Li et al. (Discrete Math 310 (2010), 3579–3583) asked how long the longest monochromatic cycle in a 2-edge-colored graph *G* with minimum degree at least could be. In this article, an answer is given for all to an asymptotic form of their question.

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*. Alex Scott (Graphs Combin 17 (2001), 539–553) proved that every connected graph *G* contains a perfect forest if and only if *G* has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP-hard, for the three others this problem is polynomial-time solvable. Moreover, every digraph with only one strong component contains a directed forest of each of these three generalization types. One of our results extends Scott's theorem to digraphs in a nontrivial way.

The article gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from *x* to *y* is equal to the complex unity *i* (and its symmetric entry is ) if the reverse arc is not present. We also allow arcs in both directions and unoriented edges, in which case we use 1 as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed—they give rise to a large number of cospectral digraphs; for every , all digraphs whose spectrum is contained in the interval are determined.

We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this article, we develop several results on 1-perfectly orientable graphs. In particular, we (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1-perfectly orientable graphs, and (iv) characterize the class of 1-perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1-perfectly orientable cobipartite graphs coincides with the class of cobipartite circular arc graphs.

We present two heuristics for finding a small power dominating set of cubic graphs. We analyze the performance of these heuristics on random cubic graphs using differential equations. In this way, we prove that the proportion of vertices in a minimum power dominating set of a random cubic graph is asymptotically almost surely at most 0.067801. We also provide a corresponding lower bound of using known results on bisection width.

In this article, we introduce and study the properties of some target graphs for 2-edge-colored homomorphism. Using these properties, we obtain in particular that the 2-edge-colored chromatic number of the class of triangle-free planar graphs is at most 50. We also show that it is at least 12.

The partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.

We prove that every normal plane map (NPM) has a path on three vertices (3-path) whose degree sequence is bounded from above by one of the following triplets: (3, 3, ∞), (3,15,3), (3,10,4), (3,8,5), (4,7,4), (5,5,7), (6,5,6), (3,4,11), (4,4,9), and (6,4,7). This description is tight in the sense that no its parameter can be improved and no term dropped. We also pose a problem of describing all tight descriptions of 3-paths in NPMs and make a modest contribution to it by showing that there exist precisely three one-term descriptions: (5, ∞, 6), (5, 10, ∞), and (10, 5, ∞).

Let *G* be a 5-connected triangulation of a surface Σ different from the sphere, and let be the Euler characteristic of Σ. Suppose that with even and *M* and *N* are two matchings in of sizes *m* and *n* respectively such that . It is shown that if the pairwise distance between any two elements of is at least five and the face-width of the embedding of *G* in Σ is at least , then there is a perfect matching *M*_{0} in containing *M* such that .

Given two graphs, a mapping between their edge-sets is *cycle-continuous*, if the preimage of every cycle is a cycle. The motivation for this definition is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph, which, if solved positively, would imply several other important conjectures (e.g., the Cycle double cover conjecture). Answering a question of DeVos, Nešetřil, and Raspaud, we prove that there exists an infinite set of graphs with no cycle-continuous mapping between them. Further extending this result, we show that every countable poset can be represented by graphs and the existence of cycle-continuous mappings between them.

The complete graph on *n* vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface *F*^{2} with Euler characteristic if and only if (resp. and ). In this article, we shall show that if quadrangulates a closed surface *F*^{2}, then has a quadrangular embedding on *F*^{2} so that the length of each closed walk in the embedding has the parity specified by any given homomorphism , called the cycle parity.

The Erdős–Lovász Tihany conjecture asserts that every graph *G* with ) contains two vertex disjoint subgraphs *G*_{1} and *G*_{2} such that and . Under the same assumption on *G*, we show that there are two vertex disjoint subgraphs *G*_{1} and *G*_{2} of *G* such that (a) and or (b) and . Here, is the chromatic number of is the clique number of *G*, and col(*G*) is the coloring number of *G*.

For a graph *G*, let denote the largest *k* such that *G* has *k* pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let denote the largest *k* such that *G* has *k* pairwise disjoint pairwise adjacent connected subgraphs of size 1 or 2. Hadwiger's conjecture states that , where is the chromatic number of *G*. Seymour conjectured for all graphs without antitriangles, that is, three pairwise nonadjacent vertices. Here we concentrate on graphs *G* with exactly one -coloring. We prove generalizations of the following statements: (i) if and *G* has exactly one -coloring then , where the proof does *not* use the four-color-theorem, and (ii) if *G* has no antitriangles and *G* has exactly one -coloring then .

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.

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.

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 .

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.

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 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.

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.

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.