L(2,1)-labelings on the modular product of two graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all non-negative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):v∈V(G)}=k. This paper considers the L(2,1)-labeling number of the modular product of two graphs and it is proved that Griggs and Yeh’s conjecture is true for the modular product of two graphs with minor exceptions.