THE L(2; 1)-CHOOSABILITY OF CYCLE

Abstract. For a given graph G = (V;E), let L(G) = fL(v) : v 2 V g be a prescribed list assignment. G is L-L(2; 1)-colorable if there exists a vertex labeling f of G such that f(v) 2 L(v) for all v 2 V ; jf(u) ?? f(v)j  2 if dG(u; v) = 1; and jf(u) ?? f(v)j  1 if dG(u; v) = 2. If G is L-L(2; 1)-colorable for every list assignment L with jL(v)j  k for all v 2 V , then G is said to be k-L(2; 1)-choosable. In this paper, we prove all cycles are 5-L(2; 1)-choosable.