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 and are not chromatic choosable. 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. 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