On dynamic coloring for planar graphs and graphs of higher genus

For integers k,r>0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G by k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v),r} different colors. The dynamic chromatic number, denoted by @g"2(G), is the smallest integer k for which a graph G has a (k,2)-coloring. A list assignment L of G is a function that assigns to every vertex v of G a set L(v) of positive integers. For a given list assignment L of G, an (L,r)-coloring of G is a proper coloring c of the vertices such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v),r} different colors and c(v)@?L(v). The dynamic choice number of G, ch"2(G), is the least integer k such that every list assignment L with |L(v)|=k, @[email protected]?V(G), permits an (L,2)-coloring. It is known that for any graph G, @g"r(G)@?ch"r(G). Using Euler distributions in this paper, we prove the following results, where (2) and (3) are best possible. (1)If G is planar, then ch"2(G)@?6. Moreover, ch"2(G)@?5 when @D(G)@?4. (2)If G is planar, then @g"2(G)@?5. (3)If G is a graph with genus g(G)>=1, then ch"2(G)@?12(7+1+48g(G)).