On L(d,1)L(d,1)-labeling of Cartesian product of a cycle and a path

A k-L(d,1)k-L(d,1)-labeling of a graph GG is a function ff from the vertex set V(G)V(G) to {0,1,…,k}{0,1,…,k} such that |f(u)−f(v)|≥1|f(u)−f(v)|≥1 if d(u,v)=2d(u,v)=2 and |f(u)−f(v)|≥d|f(u)−f(v)|≥d if d(u,v)=1d(u,v)=1. The L(d,1)L(d,1)-labeling problem is to find the L(d,1)L(d,1)-labeling number λd(G)λd(G) of a graph GG, which is the minimum cardinality kk such that GG has a k-L(d,1)k-L(d,1)-labeling. In this paper, we determine the L(d,1)L(d,1)-labeling number of the Cartesian product of a cycle and a path.