Random Minimum Length Spanning Trees in Regular Graphs

r -regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that . Received: Februray 9, 1998     

Ups and Downs of First Order Sentences on Random Graphs

whenever is a fixed positive irrational. This raises the question what zero-one valued functions on the positive irrationals arise as the limit probability of a first order sentence on these graphs. Here we prove two necessary conditions on these functions, a number-theoretic and a complexity condition. We hope to prove in a subsequent paper that these conditions together with two simpler and previously proved conditions are also sufficient and thus they constitute a characterization. Received October 2, 1998     

A Normal Law for Matchings

Dedicated to the memory of Paul Erdős, both for his pioneering discovery of normality in unexpected places, and for his questions, some of which led (eventually) to the present work.   For a simple graph G, let be the size of a matching drawn uniformly at random from the set of all matchings of G. Motivated by work of C. Godsil [11], we give, for a sequence and , several necessary and sufficient conditions for asymptotic normality of the distribution of , for instance

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

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     

The Structure of Hereditary Properties and Colourings of Random Graphs

in the probability space ? Second, does there exist a constant such that the -chromatic number of the random graph is almost surely ? The second question was posed by Scheinerman (SIAM J. Discrete Math. 5 (1992) 74–80). The two questions are closely related and, in the case p=1/2, have already been answered. Pr?mel and Steger (Contemporary Mathematics 147, Amer. Math. Soc., Providence, 1993, pp. 167-178), Alekseev (Discrete Math. Appl. 3 (1993) 191-199) and the authors ( Algorithms and Combinatorics 14 Springer-Verlag (1997) 70–78) provided an approximation which was used by the authors (Random Structures and Algorithms 6 (1995) 353–356) to answer the -chromatic question for p=1/2. However, the approximating properties that work well for p=1/2 fail completely for . In this paper we describe a class of properties that do approximate in , in the following sense: for any desired accuracy of approximation, there is a property in our class that approximates to this level of accuracy. As may be expected, our class includes the simple properties used in the case p=1/2. The main difficulty in answering the second of our two questions, that concerning the -chromatic number of , is that the number of small -graphs in has, in general, large variance. The variance is smaller if we replace by a simple approximation, but it is still not small enough. We overcome this by considering instead a very rigid non-hereditary subproperty of the approximating property; the variance of the number of small -graphs is small enough for our purpose, and the structure of is sufficiently restricted to enable us to show this by a fine analysis. Received April 20, 1999     

Ear Decompositions of Matching Covered Graphs

G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovász. We also give a simple proof of another theorem due to Lovász which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of or the fourth graph in the sequence is an odd-subdivision of . Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author. Received: November 3, 1997     

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

C _{2 k} -free subgraph of a random graph may have, obtaining best possible results for a range of p=p(n). Our estimates strengthen previous bounds of Füredi [12] and Haxell, Kohayakawa, and Łuczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and R?dl [7]. Received: February 17, 1995     

Vertex-Disjoint Triangles in Claw-Free Graphs with Minimum Degree at Least Three

. Our main result is as follows: For any integer , if G is a claw-free graph of order at least and with minimum degree at least 3, then G contains k vertex-disjoint triangles unless G is of order and G belongs to a known class of graphs. We also construct a claw-free graph with minimum degree 3 on n vertices for each such that it does not contain k vertex-disjoint triangles. We put forward a conjecture on vertex-disjoint triangles in -free graphs. Received: November 21, 1996/Revised: Revised February 19, 1998     

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     

Self-Converse and Oriented Graphs among the Third Parts of Nearly Complete Digraphs

H into t isomorphic parts is generalized so that either a remainder R or a surplus S, both of the numerically smallest possible size, are allowed. The sets of such nearly parts are defined to be the floor class and the ceiling class , respectively. We restrict ourselves to the case of nearly third parts of , the complete digraph, with . Then if , else and . The existence of nearly third parts which are oriented graphs and/or self-converse digraphs is settled in the affirmative for all or most n's. Moreover, it is proved that floor classes with distinct R's can have a common member. The corresponding result on the nearly third parts of the complete 2-fold graph is deduced. Furthermore, also if . Received: September 12, 1994/Revised: Revised November 3, 1995     

Bicliques in Graphs I: Bounds on Their Number

Bicliques are inclusion-maximal induced complete bipartite subgraphs in graphs. Upper bounds on the number of bicliques in bipartite graphs and general graphs are given. Then those classes of graphs where the number of bicliques is polynomial in the vertex number are characterized, provided the class is closed under induced subgraphs. Received January 27, 1997     

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     

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

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

Topological Subgraphs in Graphs of Large Girth

H of maximum degree , there is an integer g(H) such that every finite graph of minimum degree n and girth at least g(H) contains a subdivision of H. This had been conjectured for in [8]. We prove also that every finite 2n-connected graph of sufficiently large girth is n-linked, and this is best possible for all . Received: February 26, 1997     

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     

NOTE Spectral Characterization of Tree-Width-Two Graphs

) of a graph G, similar in spirit to his now-classical invariant . He showed that is minor-monotone and is related to the tree-width la(G) of G: and, moreover, , i.e. G is a forest. We show that and give the corresponding forbidden-minor and ear-decomposition characterizations. Received October 9, 1997/Revised July 27, 1999     

Isomorphism of Coloured Graphs with Slowly Increasing Multiplicity of Jordan Blocks

n -vertex edge coloured graphs with multiplicity of Jordan blocks bounded by k can be done in time . Received: November 29, 1994     

Exact Bounds for Judicious Partitions of Graphs

has a bipartite subgraph of size at least . We show that every graph of size has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated vertices. We also give results for partitions into more than two classes. Received: December 27, 1996/Revised: Revised June 10, 1998