The L(2,1)L(2,1)-labeling on Cartesian sum of graphs

An L(2,1)L(2,1)-labeling of a graph GG is a function ff from the vertex set V(G)V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2|f(x)−f(y)|≥2 if d(x,y)=1d(x,y)=1 and |f(x)−f(y)|≥1|f(x)−f(y)|≥1 if d(x,y)=2d(x,y)=2, where d(x,y)d(x,y) denotes the distance between xx and yy in GG. The L(2,1)L(2,1)-labeling number λ(G)λ(G) of GG is the smallest number kk such that GG has an L(2,1)L(2,1)-labeling with max{f(v):v∈V(G)}=kmax{f(v):v∈V(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ2λ(G)≤Δ2 for any simple graph with maximum degree Δ≥2Δ≥2. This paper considers the graph formed by the Cartesian sum of two graphs. As corollaries, the new graph satisfies the above conjecture (with minor exceptions).