A note on partial list coloring

Albertson, Grossman and Haas in [Discrete Math. 214 (2000), 235–240] conjecture that if L is a t-list assignment for a graph G and 1 ≤ t ≤ χ� (G), then at least t|V (G)| χ� (G) vertices of G can be colored from these lists where χ� (G) is the list chromatic number of G. In this note we investigate the partial list coloring conjecture. Precisely, we show that the conjecture is true for at least half of the numbers in the set {1, 2 ,... , χ� (G) − 1} .I n addition, we introduce a new related conjecture and finally we present some results about this conjecture. 1 Introduction and Preliminaries In this note we only consider simple graphs which are finite and undirected, with no loops or multiple edges. We mention some of the definitions which are referred to throughout the note. For a graph G, let V (G )a ndE(G) denote its vertex and edge sets, respectively. For each vertex v ∈ V (G) let L(v) be a list of allowed colors assigned to v. The collection of all lists is called a list assignment and is denoted by L . We have at-list assignment if |L(v)| = t for all v ∈ V (G). Also, we call R(L )= v∈V (G) L(v )t hecolor list of L. The graph G is called L-list colorable if there is a coloring c : V (G) → R(L )s uch that c(v) � c(u) for all uv ∈ E(G )a ndc(v) ∈L (v) for all v ∈ V (G). Moreover, G is k-choosable if it is L-list colorable for every k-list assignment L. The list chromatic number or choice number of G, denoted by χ� (G) or briefly by χ� , is the smallest number k such that G is k-choosable. List coloring was introduced independently by Vizing [7] and by Erdos, Rubin and Taylor [4]. Also, the notation λL(G) stands for the maximum number of vertices of G which are colorable with respect to the list assignment L .M oreover, setλt(G) def =m inλL(G),