Partial Online List Coloring of Graphs

For a graph G, let λt(G) be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s‐choosable and t≤s , then λt≥ts|V(G)| . In this article, we consider the online version of this conjecture. Let λt OL (G) be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s‐choosable and t≤s, then λt OL (G)≥ts|V(G)| . This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers p,q , λp OL (G)+λq OL (G)≥λp+q OL (G) . As a consequence, if s is a multiple of t, then λt OL (G)≥ts|V(G)| . We also prove that if G is online s‐choosable and t≤s−χ(G)+1 , then λt OL (G)≥ts|V(G)| and for any t≤s , λt OL (G)≥67ts|V(G)| .