Strong Chromatic Index of Sparse Graphs

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), D?bski, Grytczuk, and ?leszyńska‐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 .     

Full subgraphs

Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define Erdős, Łuczak, and Spencer proved that for , In this article, we prove the following lower bound: for , Furthermore, we show that this is tight up to a multiplicative constant factor for infinitely many p near the elements of . In contrast, we show that for any n‐vertex graph G, either G or contains a full subgraph on vertices. Finally, we discuss full subgraphs of random and pseudo‐random graphs, and several open problems.     

Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph G with colors is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree admits a cyclic interval ‐coloring if for every vertex v the degree satisfies either or . We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for ‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all ‐biregular () graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.     

Polychromatic colorings of complete graphs with respect to 1‐, 2‐factors and Hamiltonian cycles

If G is a graph and is a set of subgraphs of G, then an edge‐coloring of G is called ‐polychromatic if every graph from gets all colors present in G. The ‐polychromatic number of G, denoted , is the largest number of colors such that G has an ‐polychromatic coloring. In this article, is determined exactly when G is a complete graph and is the family of all 1‐factors. In addition is found up to an additive constant term when G is a complete graph and is the family of all 2‐factors, or the family of all Hamiltonian cycles.     

Monochromatic Kr‐Decompositions of Graphs

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic 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 monochromatic graph isomorphic to H. Let be the smallest number ? such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H‐decomposition with at most ? parts. Here, we study the function for and .     

Clique minors in double‐critical graphs

A connected t‐chromatic graph G is double‐critical if is ‐colorable for each edge . A long‐standing conjecture of Erdős and Lovász that the complete graphs are the only double‐critical t‐chromatic graphs remains open for all . Given the difficulty in settling Erdős and Lovász's conjecture and motivated by the well‐known Hadwiger's conjecture, Kawarabayashi, Pedersen, and Toft proposed a weaker conjecture that every double‐critical t‐chromatic graph contains a minor and verified their conjecture for . Albar and Gonçalves recently proved that every double‐critical 8‐chromatic graph contains a K₈ minor, and their proof is computer assisted. In this article, we prove that every double‐critical t‐chromatic graph contains a minor for all . Our proof for is shorter and computer free.     

The decycling number and maximum genus of cubic graphs

Let and denote the minimum size of a decycling set and maximum genus of a graph G, respectively. For a connected cubic graph G of order n, it is shown that . Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k‐regular graph G is , where c and are the number of components of and the number of edges in , respectively. Therefore, S is minimum if and only if is minimum. As an application, this leads to a lower bound for of a k‐regular graph G. In many cases this bound may be sharp.     

Long cycles passing through vertices in k‐connected graphs

In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k‐connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values of m and k, with , must G have a cycle of length at least passing through X? Fujisawa and Yamashita solved this problem for the case and in 2008. We provide an affirmative answer to this problem for the case of and .     

Dominating sets inducing large components in maximal outerplanar graphs

For a maximal outerplanar graph G of order n at least three, Matheson and Tarjan showed that G has domination number at most . Similarly, for a maximal outerplanar graph G of order n at least five, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that G has total domination number at most unless G is isomorphic to one of two exceptional graphs of order 12. We present a unified proof of a common generalization of these two results. For every positive integer k, we specify a set of graphs of order at least and at most such that every maximal outerplanar graph G of order n at least that does not belong to has a dominating set D of order at most such that every component of the subgraph of G induced by D has order at least k.     

A relaxation of the strong Bordeaux Conjecture

Let be k nonnegative integers. A graph G is ‐colorable if the vertex set can be partitioned into k sets , such that the subgraph , induced by , has maximum degree at most for . Let denote the family of plane graphs with neither adjacent 3‐cycles nor 5‐cycles. Borodin and Raspaud (2003) conjectured that each graph in is (0, 0, 0)‐colorable (which was disproved very recently). In this article, we prove that each graph in is (1, 1, 0)‐colorable, which improves the results by Xu (2009) and Liu‐Li‐Yu (2016).     

On the Structure of Graphs with Given Odd Girth and Large Minimum Degree

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd?s, and Sós implies that every n‐vertex graph with odd girth and minimum degree bigger than must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases and 3, we show that every n‐vertex graph with odd girth and minimum degree bigger than is homomorphic to the cycle of length . This is best possible in the sense that there are graphs with minimum degree and odd girth that are not homomorphic to the cycle of length . Similar results were obtained by Brandt and Ribe‐Baumann.     

Low Polynomial Exclusion of Planar Graph Patterns

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 .     

Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .     

List Coloring with a Bounded Palette

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 ? 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 .     

Partition functions and a generalized coloring‐flow duality for embedded graphs

Let G be a finite group and a class function. Let be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection F of faces of H. Define the partition function , where denotes the product of the κ‐values of the edges incident with v (in cyclic order), where the inverse is taken for edges leaving v. Write , where the sum runs over irreducible representations λ of G with character and with for every λ. When H is connected, it is proved that , where 1 is the identity element of G. Among the corollaries, a formula for the number of nowhere‐identity G‐flows on H is derived, generalizing a result of Tutte. We show that these flows correspond bijectively to certain proper G‐colorings of a covering graph of the dual graph of H. This correspondence generalizes coloring‐flow duality for planar graphs.     

Nowhere‐zero 3‐flow and ‐connectedness in graphs with four edge‐disjoint spanning trees

Given a zero‐sum function with , an orientation D of G with in for every vertex is called a β‐orientation. A graph G is ‐connected if G admits a β‐orientation for every zero‐sum function β. Jaeger et al. conjectured that every 5‐edge‐connected graph is ‐connected. A graph is ‐extendable at vertex v if any preorientation at v can be extended to a β‐orientation of G for any zero‐sum function β. We observe that if every 5‐edge‐connected essentially 6‐edge‐connected graph is ‐extendable at any degree five vertex, then the above‐mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lovász et al., we prove that every graph with at least four edge‐disjoint spanning trees is ‐connected. Consequently, every 5‐edge‐connected essentially 23‐edge‐connected graph is ‐extendable at any degree five vertex.     

More on the Bipartite Decomposition of Random 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 . Erd?s 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.     

Gomory‐Hu trees of infinite graphs with finite total weight

A well‐known theorem of Gomory and Hu states that if G is a finite graph with nonnegative weights on its edges, then there exists a tree T (now called a Gomory‐Hu tree) on such that for all there is an such that the two components of determine an optimal (minimal valued) cut between u an v in G. In this article, we extend their result to infinite weighted graphs with finite total weight. Furthermore, we show by an example that one cannot omit the condition of the finiteness of the total weight.     

Plane Graphs with Maximum Degree Are Entirely ‐Colorable

Let be a plane graph with the sets of vertices, edges, and faces V, E, and F, respectively. If one can color all elements in using k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k‐colorable. Kronk and Mitchem [Discrete Math 5 (1973) 253‐260] conjectured that every plane graph with maximum degree Δ is entirely ‐colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490–501), where the authors asked: is every simple plane graph entirely ‐colorable? In this article, we prove that every simple plane graph with is entirely ‐colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely ‐colorable.     

Forbidden Pairs of Disconnected Graphs Implying Hamiltonicity

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 n₀ such that every 2‐connected ‐free graph of order at least n₀ is hamiltonian.