Perfect Matchings in Balanced Hypergraphs – A Combinatorial Approach

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuéjols, Kapoor, and Vušković extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof. Received April 29, 1997     

Density Conditions for Panchromatic Colourings of Hypergraphs

Let be a hypergraph. A panchromatic t-colouring of is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of is . Among other results, it is proved here that if every edge has at least t vertices and whenever , then is panchromatically t-choosable, and this condition is sharp; the minimum such that every t-uniform hypergraph with is panchromatically t-choosable satisfies ; and except possibly when t = 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if whenever , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree. Received November 10, 1998 RID="*" ID="*" This work was carried out while the first author was visiting Nottingham, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.     

Decomposing Hypergraphs into Simple Hypertrees

T be a simple k-uniform hypertree with t edges. It is shown that if H is any k-uniform hypergraph with n vertices and with minimum degree at least , and the number of edges of H is a multiple of t then H has a T-decomposition. This result is asymptotically best possible for all simple hypertrees with at least two edges. Received December 28, 1998     

Connected [a,b]-factors in Graphs

In this paper, we prove the following result: Let G be a connected graph of order n, and minimum degree . Let a and b two integers such that 2a <= b. Suppose and . Then G has a connected [a,b]-factor. Received February 10, 1998/Revised July 31, 2000     

Perfect Graphs, Partitionable Graphs and Cutsets

We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous pairs. Received December 13, 1999 RID="*" ID="*" This work was supported in part by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, and by NSF grants DMI-0098427 and DMI-9802773 and ONR grant N00014-97-1-0196.     

Totally Odd -subdivisions in 4-chromatic Graphs

We prove the conjecture made by Bjarne Toft in 1975 that every 4-chromatic graph contains a subdivision of in which each edge of corresponds to a path of odd length. As an auxiliary result we characterize completely the subspace of the cycle space generated by all cycles through two fixed edges. Toft's conjecture was proved independently in 1995 by Wenan Zang. Received May 26, 1998     

Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

Small Universal Graphs for Bounded-Degree Planar Graphs

For all positive integers N and k, let denote the family of planar graphs on N or fewer vertices, and with maximum degree k. For all positive integers N and k, we construct a -universal graph of size . This construction answers with an explicit construction the previously open question of the existence of such a graph. Received July 8, 1998 RID="*" ID="*" Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

On a Max-min Problem Concerning Weights of Edges

The weight w(e) of an edge e = uv of a graph is defined to be the sum of degrees of the vertices u and v. In 1990 P. Erdős asked the question: What is the minimum weight of an edge of a graph G having n vertices and m edges? This paper brings a precise answer to the above question of Erdős. Received July 12, 1999     

A Tree Version of Kőnig's Theorem

Kőnig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed. Received January 27, 2000/Revised November 2, 2000 RID=" " ID=" " The research of the first author was supported by grants from the Israel Science Foundation, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.     

Homomorphisms of Products of Graphs into Graphs Without Four Cycles

Given two graphs A and G, we write if there is a homomorphism of A to G and if there is no such homomorphism. The graph G is -free if, whenever both a and c are adjacent to b and d, then a = c or b = d. We will prove that if A and B are connected graphs, each containing a triangle and if G is a -free graph with and , then (here " denotes the categorical product). Received August 31, 1998/Revised April 19, 2000 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.     

Size Ramsey Numbers of Stars Versus 3-chromatic Graphs

  Let be the star with n edges, be the triangle, and be the family of odd cycles. We establish the following bounds on the corresponding size Ramsey numbers.

Graph Orientations with Edge-connection and Parity Constraints

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with |V| + |E| even has a (k-1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge-disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s. Received December 14, 1998/Revised January 12, 2001 RID="*" ID="*" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772. Part of research was done while this author was visiting EPFL, Lausanne, June, 1998. RID="†" ID="†" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772 and OTKA T030059.     

Integer and Fractional Packings in Dense Graphs

Let be any fixed graph. For a graph G we define to be the maximum size of a set of pairwise edge-disjoint copies of in G. We say a function from the set of copies of in G to [0, 1] is a fractional -packing of G if for every edge e of G. Then is defined to be the maximum value of over all fractional -packings of G. We show that for all graphs G. Received July 27, 1998 / Revised December 3, 1999     

The Distance-regular Graphs with Intersection Number and with an Eigenvalue

and with an eigenvalue . Received: October 2, 1995/Revised: Revised November 26, 1997     

Ramsey Numbers for Trees of Small Maximum Degree

For a tree T we write and , , for the sizes of the vertex classes of T as a bipartite graph. It is shown that for T with maximum degree , the obvious lower bound for the Ramsey number R(T,T) of is asymptotically the correct value for R(T,T). Received December 15, 1999 RID=" " ID=" " The first and third authors were partially supported by NSERC. The second author was partially supported by KBN grant 2 P03A 021 17.     

NOTE The Johnson Graphs Satisfy a Distance Extension Property

G has property if whenever F and H are connected graphs with and |H|=|F|+1, and and are isometric embeddings, then there is an isometric embedding such that . It is easy to construct an infinite graph with for all k, and holds in almost all finite graphs. Prior to this work, it was not known whether there exist any finite graphs with . We show that the Johnson graphs J(n,3) satisfy whenever , and that J(6,3) is the smallest graph satisfying . We also construct finite graphs satisfying and local versions of the extension axioms studied in connection with the Rado universal graph. Received June 9, 1998     

Subdivisions of a Graph of Maximal Degree n + 1 in Graphs of Average Degree and Large Girth

Dedicated to the memory of Paul Erdős It is proved that for every finite graph H of maximal degree and every , there is an integer such that every finite graph of average degree at least and of girth at least contains a subdivision of H. Received May 5, 1999     

The Erdős–Pósa Property for Odd Cycles in Graphs of Large Connectivity

Dedicated to the memory of Paul Erdős A graph G is k-linked if G has at least 2k vertices, and, for any vertices , , ..., , , , ..., , G contains k pairwise disjoint paths such that joins for i = 1, 2, ..., k. We say that G is k-parity-linked if G is k-linked and, in addition, the paths can be chosen such that the parities of their lengths are prescribed. We prove the existence of a function g(k) such that every g(k)-connected graph is k-parity-linked if the deletion of any set of less than 4k-3 vertices leaves a nonbipartite graph. As a consequence, we obtain a result of Erdős–Pósa type for odd cycles in graphs of large connectivity. Also, every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph. Received May 13, 1999/Revised June 19, 2000