Bounds on L(2,1)-choice number of Cartesian products of paths and spiders

For a given graph G = (V,E), let L(G) = {L(v) : v ∈ V } be a prescribed list assignment. G is L-L(2, 1)-colorable if there exists a vertex labeling f of G such that [function of](v) ∈ L(v) for all v ∈ V ; |[function of](u)-[function of](v)| ≥ 2 if d^sub G^(u, v) = 1; and |[function of](u)-[function of](v)| ≥ 1 if d^sub G^(u, v) = 2. If G is L-L(2, 1)-colorable for every list assignment L with |L(v)| ≥ k for all v ∈ V , then G is said to be k-L(2, 1)-choosable. The L(2, 1)-choice number λl(G) of a graph G is the smallest number k such that G is k-L(2, 1)-choosable. In this paper, we provide a lower bound on λl for the Cartesian products of a path and a spider. Also we provide an upper bound on λl for the Cartesian products of K^sub 2^ and a spider. The L(2, 1)-choice number of spider is determined.