Dynamic list coloring of bipartite graphs

A dynamic coloring of a graph is a proper coloring of its vertices such that every vertex of degree more than one has at least two neighbors with distinct colors. The least number of colors in a dynamic coloring of G, denoted by χ2(G), is called the dynamic chromatic number of G. The least integer k, such that if every vertex of G is assigned a list of k colors, then G has a proper (resp. dynamic) coloring in which every vertex receives a color from its own list, is called the choice number of G, denoted ch(G) (resp. the dynamic choice number, denoted ch2(G)). It was recently conjectured [S. Akbari et al., On the list dynamic coloring of graphs, Discrete Appl. Math. (2009)] that for any graph G, ch2(G) = max(ch(G), χ2(G)). In this short note we disprove this conjecture. We first give the example of a small planar bipartite graph G with ch(G) = χ2(G) = 3 and ch2(G) = 4. Then, for any integer k ≥ 5, we construct a bipartite graph Gk such that ch(Gk) = χ2(Gk) = 3 and ch2(G) ≥ k.