On Friday, 26 March 1993, Paul Erdős celebrated his 80th birthday. To honour him on this occasion, a conference was held in Trinity College, Cambridge, under the auspices of the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge. Many of the world's best combinatorialists came to pay tribute to Erdős, the universally acknowledged leader of their field.

The conference was generously supported both by the London Mathematical Society and by the Heilbronn Fund of Trinity College. As at former Cambridge Conferences in honour of Paul Erdős, the day-to-day running of this conference was in the able hands of Gabriella Bollobás, with the untiring assistance of Tristan Denley, Ted Dobson, Tom Gamblin, Chris Jagger, Imre Leader, Alex Scott and Alan Stacey. The conference would not have taken place without their dedicated work.

On the eve of Erdős' birthday, a sumptuous feast was held in his honour in the Hall of Trinity College. The words wherein he was toasted are reproduced in the following pages. This volume of research papers was presented to Paul Erdős by its authors as their own toast, gladly offered with their gratitude respect and warmest wishes.

Sadly, before this book reached its printed form, Paul Erdős died. Whereas it was conceived in joy it appears now tinged with sorrow. We feel his loss tremendously. But it is not appropriate that grief should overshadow this volume.

Introduction and notation

Whereas the characterization of all graphs having the property that the deletion of any two edges decreases the connectivity number by two is rather easy, and well known (see Section 2), the characterization of all graphs with the analogous property for the deletion of two vertices instead of two edges seems to be hopeless. So the following idea suggests itself. A graph or digraph G is called vertex-edge-critically n-connected (abbreviated to n-ve-critical), if the deletion of any vertex v and any edge e not incident to v decreases the connectivity number n of G by two (and such v and e exist). If we do not want to specify the connectivity number, we write vertex-edge-critical or ve-critical. When I determined the minimum number of 1-factors of a (2k)-connected graph containing a 1-factor, the ve-critical graphs played an important role and all ve-critical undirected graphs were characterized there. It was shown in that every ve-critical undirected graph is obtained in the following way. For an integer m ≥ 1, take vertex-disjoint circuits of length m + 2 and vertex-disjoint copies of (the complementary graph of the complete graph Km on m vertices) and take all edges between these vertex-disjoint graphs.

Introduction

Dirac's theorem states that every graph G with minimum degree δ(G) ≥ |G|/2 has a hamilton cycle. The simplest analogue for digraphs is given by the theorem of Ghouila-Houri. Given a digraph G of order n and a vertex v ∈ G, we denote the outdegree of v by d+(v) and the indegree by d−(v). We also define d°(v) to be min{d+(v), d−(v)}, and δ°(G) to be min{d°(v): v ∈ G}. Ghouila-Houri's theorem implies that G contains a directed hamilton cycle if d°(G)(G) ≥ n/2. Only recently has a constant c < ½ been established such that every oriented graph satisfying δ°(G) > cn has a directed hamilton cycle; Häggkvist has shown that c = (½ − 2−15) will suffice. He also showed that the condition δ°(G) > n/3 proposed by Thomassen is inadequate to guarantee a hamilton cycle, and conjectured that δ°(G) ≥ 3n/8 is sufficient.

When considering hamilton cycles in digraphs there is no reason to stick to directed cycles only; we might ask for any orientation of an n-cycle. For tournaments G, Thomason has shown that G will contain every oriented cycle (except the directed cycle if G is not strong) regardless of the degrees, provided n is large.