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Keywords:

  • choice number;
  • online choice number;
  • partial list coloring;
  • online partial list coloring

Abstract

For a graph G, let inline image 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 inline image, then inline image. In this article, we consider the online version of this conjecture. Let inline image 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 inline image then inline image. This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers inline image, inline image. As a consequence, if s is a multiple of t, then inline image. We also prove that if G is online s-choosable and inline image, then inline image and for any inline image, inline image.