Dynamic list coloring of bipartite graphs

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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 2(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), 2(G)). In this short note we disprove this conjecture. We first give the example of a small planar bipartite graph G with ch(G) = 2(G) = 3 and ch2(G) = 4. Then, for any integer k 5, we construct a bipartite graph Gk such that ch(Gk) = 2(Gk) = 3 and ch2(G) k.
Louis Esperet
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2010
Where DAM
Authors Louis Esperet
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