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 .     

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.     

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.     

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 .     

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 .     

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.     

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

For a finite set V and a positive integer k with , letting be the set of all k‐subsets of V, the pair is called the complete k‐hypergraph on V, while each k‐subset of V is called an edge. A factorization of the complete k‐hypergraph of index , simply a ‐factorization of order n, is a partition of the edges into s disjoint subsets such that each k‐hypergraph , called a factor, is a spanning subhypergraph of . Such a factorization is homogeneous if there exist two transitive subgroups G and M of the symmetric group of degree n such that G induces a transitive action on the set and M lies in the kernel of this action. In this article, we give a classification of homogeneous factorizations of that admit a group acting transitively on the edges of . It is shown that, for and , there exists an edge‐transitive homogeneous ‐factorization of order n if and only if is one of (32, 3, 5), (32, 3, 31), (33, 4, 5), , and , where and q is a prime power with .     

Decompositions of Graphs into Fans and Single Edges

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 ? such that any graph G of order n admits an H‐decomposition with at most ? 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.     

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.     

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.     

Petersen Cores and the Oddness of Cubic Graphs

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 .     

Cycle‐Saturated Graphs with Minimum Number of Edges

A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by . We prove holds for all , where is a cycle with length k. A graph G is H‐semisaturated if contains more copies of H than G does for . Let be the minimum size of an n‐vertex H‐semisaturated graph. We have We conjecture that our constructions are optimal for . © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013     

Extensions of Fractional Precolorings Show Discontinuous Behavior

We study the following problem: given a real number k and an integer d, what is the smallest ε such that any fractional ‐precoloring of vertices at pairwise distances at least d of a fractionally k‐colorable graph can be extended to a fractional ‐coloring of the whole graph? The exact values of ε were known for and any d. We determine the exact values of ε for if , and if , and give upper bounds for if , and if . Surprisingly, ε viewed as a function of k is discontinuous for all those values of d.     

On Hamilton decompositions of infinite circulant graphs

The natural infinite analog of a (finite) Hamilton cycle is a two‐way‐infinite Hamilton path (connected spanning 2‐valent subgraph). Although it is known that every connected 2k‐valent infinite circulant graph has a two‐way‐infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge‐disjoint two‐way‐infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k‐valent connected circulant graph has a decomposition into k edge‐disjoint Hamilton cycles. We settle the problem of decomposing 2k‐valent infinite circulant graphs into k edge‐disjoint two‐way‐infinite Hamilton paths for , in many cases when , and in many other cases including where the connection set is or .     

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.     

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.     

Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs

For a graph H , let for every edge . For and , let be a set of k‐edge‐connected K₃‐free graphs of order at most r and without spanning closed trails. We show that for given and ε, if H is a k‐connected claw‐free graph of order n with and , and if n is sufficiently large, then either H is Hamiltonian or the Ryjác?ek's closure where G is an essentially k‐edge‐connected K₃‐free graph that can be contracted to a graph in . As applications, we prove:

• (i) For , if and if and n is sufficiently large, then H is Hamiltonian.
• (ii) For , if and n is sufficiently large, then H is Hamiltonian.

These bounds are sharp. Furthermore, since the graphs in are fixed for given p and can be determined in a constant time, any improvement to (i) or (ii) by increasing the value of p and so enlarging the number of exceptions can be obtained computationally.     

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).     

Equimatchable Graphs on Surfaces

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