Counterexamples to the List Square Coloring Conjecture

A graph G is called chromatic-choosable if χl(G) = χ(G). It is an interesting problem to find graphs that are chromatic-choosable. There are several famous conjectures that some classes of graphs are chromatic-choosable including the List Coloring Conjecture, which states that any line graph is chromatic-choosable. The square G of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G 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. Kostochka and Woodall [4] proposed the following conjecture, which is called List Square Coloring Conjecture.