L(2,1)-labeling of interval graphs

An $$L(2,1)$$L(2,1)-labeling of a graph $$G=(V,E)$$G=(V,E) is a function $$f$$f from the vertex set $$V(G)$$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 $$L(2,1)$$L(2,1)-labeling number denoted by $$\lambda _{2,1}(G)$$λ2,1(G) of $$G$$G is the minimum range of labels over all such labeling. In this article, it is shown that, for an interval graph $$G$$G, the upper bound of $$\lambda _{2,1}(G)$$λ2,1(G) is $$\Delta +\omega $$Δ+ω, where $$\Delta $$Δ and $$\omega $$ω represents the maximum degree of the vertices and size of maximum clique respectively. An $$O(m+n)$$O(m+n) time algorithm is also designed to $$L(2,1)$$L(2,1)-label a connected interval graph, where $$m$$m and $$n$$n represent the number of edges and vertices respectively. Extending this idea it is shown that $$\lambda _{2,1}(G)\le \Delta +3\omega $$λ2,1(G)≤Δ+3ω for circular-arc graph.