Algorithmic Aspects Of Partial List Colourings

Received Let G = (V; E) be a graph with n vertices, chromatic number (G) and list chromatic number`(G): Suppose each vertex of V (G) is assigned a list of t colors. Albertson, Grossman and Haas 1] conjectured that at least t ` (G) n vertices can be colored properly from these lists. Albertson et. al. 1] and Chappell 3] proved partial results concerning this conjecture. This paper presents algorithms which color at least the number of vertices given in the bounds of Albertson et. al. and Chappell. Especially it follows that the conjecture is valid for all bipartite graphs and that for every bipartite graph and every assignment of lists with t colors in each list where 0 t ` (G) it is possible to color at least (1 ? (1 2) t)n vertices in polynomial time. Thus, if G is bipartite and L is a list assignment with jL(v)j log 2 n for all v 2 V then G is L-list colorable in polynomial time. 1. Introduction Let G be a graph with vertex set V , jV j = n, edge set E and chromatic number (G). Furthermore let L(v) be a list of allowed colors assigned to each vertex v 2 V (G). The collection of all lists is called a list assignment and denoted by L. The graph G is called L-list colorable if there is a coloring c of the vertices of G such that c(v) 6 = c(w) for all vw 2 E(G) and c(v) 2 L(v) for all v 2 V (G). Furthermore, G is k-choosable if it is L-list colorable for every list assignment L with jL(v)j = k for all v 2 V (G). The list chromatic number`(G) is the smallest number k such that G is k ? choosable.