The List Square Coloring Conjecture fails for cubic bipartite graphs and planar line graphs

Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartie graphs and alternatively they developed their construction to bipartite graphs such that one partite set has maximum degree $7$. Motivated by the List Total Coloring Conjecture, they also asked whether this number can be pushed down to $2$. At about the same time, Kim, SooKwon, and Park (2015) asked whether there would exist a claw-free counterexample to establish a generalization for a conjecture of Gravier and Maffray (1997). In this note, we answer the problem of Kim and Park by pushing the desired upper bound down to $3$ by introducing a family of cubic bipartite counterexamples, and positively answer the problem of Kim, SooKwon, and Park by introducing a family of planar line graphs