A question recently posed by Häggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

Let G = (V, E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χ[lscr ](G). Suppose each vertex of V(G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least [formula here] vertices can be coloured properly from these lists.

Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 [les ] t [les ] χ[lscr ](G), it is possible to colour at least (1 − (1/2)t)n vertices in polynomial time. Thus, if G is bipartite and [Lscr ] is a list assignment with [mid ]L(v)[mid ] [ges ] log2n for all v ∈ V, then G is [Lscr ]-list colourable in polynomial time.