A survey on labeling graphs with a condition at distance two Original Research Article

For positive integers k,d1,d2k,d1,d2, a k-L(d1,d2)L(d1,d2)-labeling of a graph G is a function f:V(G)→{0,1,2,…,k}f:V(G)→{0,1,2,…,k} such that |f(u)-f(v)|⩾di|f(u)-f(v)|⩾di whenever the distance between u and vv is i in G, for i=1,2i=1,2. The L(d1,d2)L(d1,d2)-number of G, λd1,d2(G)λd1,d2(G), is the smallest k such that there exists a k-L(d1,d2)L(d1,d2)-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.