Nonseparating Cycles Avoiding Specific Vertices

Thomassen proved that every ‐connected graph G contains an induced cycle C such that is k‐connected, establishing a conjecture of Lovász. In general, one could ask the following question: For any positive integers , does there exist a smallest positive integer such that for any ‐connected graph G, any with , and any , there is an induced cycle C in such that and is l‐connected? The case when is a well‐known conjecture of Lovász that is still open for . In this article, we prove and . We also consider a weaker version: For any positive integers , is there a smallest positive integer such that for every ‐connected graph G and any with , there is an induced cycle C in such that is l‐connected? The case when was studied by Thomassen. We prove and .     

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.     

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.     

Complete graph immersions and minimum degree

An immersion of a graph H in another graph G is a one‐to‐one mapping and a collection of edge‐disjoint paths in G, one for each edge of H, such that the path corresponding to the edge has endpoints and . The immersion is strong if the paths are internally disjoint from . We prove that every simple graph of minimum degree at least contains a strong immersion of the complete graph . This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al. Our result supports a conjecture of Lescure and Meyniel (also independently proposed by Abu‐Khzam and Langston), which is the analogue of famous Hadwiger’s conjecture for immersions and says that every graph without a ‐immersion is ‐colorable.     

A Density Turán Theorem

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 ?‐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.     

More on foxes

An edge in a k‐connected graph G is called k‐contractible if the graph obtained from G by contracting e is k‐connected. Generalizing earlier results on 3‐contractible edges in spanning trees of 3‐connected graphs, we prove that (except for the graphs if ) (a) every spanning tree of a k‐connected triangle free graph has two k‐contractible edges, (b) every spanning tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (c) for , every DFS tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (d) every spanning tree of a cubic 3‐connected graph nonisomorphic to K₄ has at least many 3‐contractible edges, and (e) every DFS tree of a 3‐connected graph nonisomorphic to K₄, the prism, or the prism plus a single edge has two 3‐contractible edges. We also discuss in which sense these theorems are best possible.     

On a Problem of Judicious k‐Partitions of Graphs

For positive integers and m , let be the smallest integer such that for each graph G with m edges there exists a k‐partition in which each contains at most edges. Bollobás and Scott showed that . Ma and Yu posed the following problem: is it true that the limsup of tends to infinity as m tends to infinity? They showed it holds when k is even, establishing a conjecture of Bollobás and Scott. In this article, we solve the problem completely. We also present a result by showing that every graph with a large k‐cut has a k‐partition in which each vertex class contains relatively few edges, which partly improves a result given by Bollobás and Scott.     

Smallest graphs with given generalized quaternion automorphism group

For , a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If , then such a graph has vertices and edges. In the special case when , a smallest graph has 16 vertices but 44 edges.     

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 a Conjecture of Erdős,Gallai, and Tuza

Erd?s, Gallai, and Tuza posed the following problem: given an n‐vertex graph G, let denote the smallest size of a set of edges whose deletion makes G triangle‐free, and let denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that ? We have two main results. We first obtain the upper bound , as a partial result toward the Erd?s–Gallai–Tuza conjecture. We also show that always , where m is the number of edges in G; this bound is sharp in several notable cases.     

Thoroughly dispersed colorings

We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly dispersed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define as the maximum number of colors in such a coloring and as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least  two has  and this is best possible. For planar graphs, we show that every triangular disc has and this is best possible, and that every planar graph has and this is best possible, while we conjecture that every planar triangulation has . Further, although there are arbitrarily large examples of connected, cubic graphs with , we show that for a connected cubic graph . We also consider the related concepts in hypergraphs.     

Large Induced Forests in Graphs

In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least . The second is that every connected triangle‐free graph of order n and size m has an induced forest of order at least . This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound of Alon–Mubayi–Thomas for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least . The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least , where was conjectured to be the best lower bound by Kowalik, Lu?ar, and ?krekovski.     

Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles

According to the classical Erd?s–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex‐disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in or there exists a set S of at most vertices such that contains no subgraph isomorphic to any graph in . However, the function f is exponential. In this note, we prove that this function becomes quadratic when consists all graphs that can be contracted to a fixed planar graph . For a fixed c, is the graph with two vertices and parallel edges. Observe that for this corresponds to the classical Erd?s–Pósa theorem.     

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.     

Edge‐Disjoint Spanning Trees,Edge Connectivity,and Eigenvalues in Graphs

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

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

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

Two Questions of Erdős on Hypergraphs above the Turán Threshold

For ordinary graphs it is known that any graph G with more edges than the Turán number of must contain several copies of , and a copy of , the complete graph on vertices with one missing edge. Erd?s asked if the same result is true for , the complete 3‐uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for . For the second property, that of containing a , we show that for the answer is negative for all large n as well, by proving that the Turán density of is greater than that of .     

Edge‐colorings avoiding rainbow stars

We consider an extremal problem motivated by a article of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge‐colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given , we look for n‐vertex graphs that admit the maximum number of r‐edge‐colorings such that at most colors appear in edges incident with each vertex, that is, r‐edge‐colorings avoiding rainbow‐colored stars with t edges. For large n, we show that, with the exception of the case , the complete graph is always the unique extremal graph. We also consider generalizations of this problem.     

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.