A survey on labeling graphs with a condition at distance two

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.