On the Choice Numbers of Some Complete Multipartite Graphs

Abstract

For every vertex v in a graph G, let L(v) denote a list of colors assigned to v. A list coloring is a proper coloring f such that f(v) ∈ L(v) for all v. A graph is k-choosable if it admits a list coloring for every list assignment L with |L(v)| = k. The choice number of G is the minimum k such that G is k-choosable. We generalize a result (of [4]) concerning the choice numbers of complete bipartite graphs and prove some uniqueness results concerning the list colorability of the complete (k + 1)-partite graph K _{n, ..., n, m}. Using these, we determine the choice numbers for some complete multipartite graphs K _{n, ..., n, m}. As a byproduct, we classify (i) completely those complete tripartite graphs K _{2, 2, m} and (ii) almost completely those complete bipartite graphs K _{n, m} (for n ≤ 6) according to their choice numbers.