Algorithmic Aspects of Partial List Colourings

Abstract

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.