A survey on labeling graphs with a condition at distance two

Abstract

For positive integers k, d1, d2, a k- L ( d1, d2 )-labeling of a graph G is a function f : V ( G ) → { 0, 1, 2, ..., k } such that | f ( u ) - f ( v ) | {greater than or slanted equal to} di whenever the distance between u and v is i in G, for i = 1, 2. The L ( d1, d2 )-number of G, λd1 ,d2 ( G ), is the smallest k such that there exists a k- 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. © 2006 Elsevier B.V. All rights reserved.