We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvátal's well-known degree condition for hamiltonicity is best possible.

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t -tough. We first give a best monotone theorem when t ≥ 1, but then show that for any integer k ≥ * Supported by EPSRC grant EP/F064551/1.

One of Frank Boesch's best known papers is 'The strongest monotone degree condition for n-connectedness of a graph' (Boesch, J Combinatorial Theory Ser B 16 (1974), 162-165.). In this article, we give a simple sufficient degree condition for a graph to be k -edgeconnected, and also give the strongest monotone condition for a graph to be 2-edge-connected.

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V (G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.

a b s t r a c tWe give sufficient conditions on the vertex degrees of a graph G to guarantee that G has binding number at least b, for any given b > 0. Our conditions are best possible in exactly the same way that Chvátal's well-known degree condition to guarantee a graph is Hamiltonian is best possible.