On some results for the L(2,1)-labeling on Cartesian sum 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 nonnegative integers such that vertical bar f (x) - f (y)vertical bar >= 2 if d(x, y) = 1 and vertical bar f (x) - f(y)vertical bar >= 1 if d(x, y) = 2, where d(x, y) denotes the distance between vertices x and y in G. The L(2, 1)-labeling number lambda(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f(v) : v is an element of V(G)} = k. We consider Cartesian sums of graphs and derive, both, lower and upper bounds for the L(2, 1)-labeling number of this class of graphs; we use two approaches to derive the upper bounds for the L(2, 1)-labeling number and both approaches improve previously known upper bounds. We also present several approximation algorithms for computing L(2, 1)-labelings for Cartesian sum graphs.