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.     

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 .     

Smallest domination number and largest independence number of graphs and forests with given degree sequence

For a sequence d of nonnegative integers, let and be the sets of all graphs and forests with degree sequence d, respectively. Let , , , and where is the domination number and is the independence number of a graph G. Adapting results of Havel and Hakimi, Rao showed in 1979 that can be determined in polynomial time. We establish the existence of realizations with , and with and that have strong structural properties. This leads to an efficient algorithm to determine for every given degree sequence d with bounded entries as well as closed formulas for and .     

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 .     

Disjoint perfect matchings in 3‐uniform hypergraphs

For a hypergraph H, let denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with , if H is a 3‐uniform hypergraph with order n and then H has a perfect matching, and this bound on is best possible. In this article, we show that under the same conditions, H contains at least pairwise disjoint perfect matchings, and this bound is sharp.     

Ramsey Results for Cycle Spectra

Let denote the set of lengths of cycles of a graph G of order n and let denote the complement of G. We show that if , then contains all odd ? with and all even ? with , where and denote the maximum odd and the maximum even integer in , respectively. From this we deduce that the set contains at least integers, which is sharp.     

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.     

Sufficient conditions for the existence of a path‐factor which are related to odd components

In this article, we are concerned with sufficient conditions for the existence of a ‐factor. We prove that for , there exists such that if a graph G satisfies for all , then G has a ‐factor, where is the number of components C of with . On the other hand, we construct infinitely many graphs G having no ‐factor such that for all .     

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 .     

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 .     

Packing Countably Many Branchings with Prescribed Root‐Sets in Infinite Digraphs

We generalize an unpublished result of C. Thomassen. Let be a digraph and let be a multiset of subsets of V in such a way that any backward‐infinite path in D meets all the sets . We show that if all is simultaneously reachable from the sets by edge‐disjoint paths, then there exists a system of edge‐disjoint spanning branchings in D where the root‐set of is .     

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.     

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.     

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.     

Packing odd T‐joins with at most two terminals

Take a graph G, an edge subset , and a set of terminals where is even. The triple is called a signed graft. A T‐join is odd if it contains an odd number of edges from Σ. Let ν be the maximum number of edge‐disjoint odd T‐joins. A signature is a set of the form where and is even. Let τ be the minimum cardinality a T‐cut or a signature can achieve. Then and we say that packs if equality holds here. We prove that packs if the signed graft is Eulerian and it excludes two special nonpacking minors. Our result confirms the Cycling Conjecture for the class of clutters of odd T‐joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two‐commodity paths, packing T‐joins with at most four terminals, and a new result on covering edges with cuts.     

A Relaxed Version of the Erdős–Lovász Tihany Conjecture

The Erd?s–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 .     

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.     

Calculating Ramsey Numbers by Partitioning Colored Graphs

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 Erd?s 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.