L(2,1)-labeling of interval graphs

An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The L(2,1)- labeling number denoted by λ2,1(G) of G is the minimum range of labels over all such labeling. In this article, it is shown that, for an interval graph G, the upper bound of λ2,1(G)is � +ω,whereand ωrepresentsthemaximumdegreeoftheverticesand size of maximum clique respectively. An O(m + n) time algorithm is also designed to L(2,1)-label a connected interval graph, where m and n represent the number of edges and vertices respectively. Extending this idea it is shown that λ2,1(G) ≤ � +3ω for circular-arc graph.