M ay 2 01 4 Bipartite graphs whose squares are not chromatic - choosable

The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let χ(H) and χl(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χl(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that G2 is chromatic-choosable for every graph G. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether G2 is chromatic-choosable or not for every bipartite graph G. In this paper, we give a bipartite graph G such that χl(G 2) 6= χ(G2). Moreover, we show that the value χl(G 2)− χ(G2) can be arbitrarily large.