L(2,1)-labelling of Circular-arc Graph

TL;DR

This paper establishes an upper bound for the L(2,1)-labelling number of circular-arc graphs, relating it to maximum degree and clique size, contributing to graph labelling theory.

Contribution

It provides a new upper bound for the L(2,1)-labelling number specifically for circular-arc graphs, linking it to key graph parameters.

Findings

Upper bound of λ_{2,1}(G) is Δ+3ω for circular-arc graphs.

The bound relates the labelling number to maximum degree and maximum clique size.

Advances understanding of graph labelling constraints for circular-arc graphs.

Abstract

An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ 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)-labelling number denoted by $\lambda_{2,1}(G)$ of $G$ is the minimum range of labels over all such labelling. In this article, it is shown that, for a circular-arc graph $G$, the upper bound of $\lambda_{2,1}(G)$ is $\Delta+3\omega$, where $\Delta$ and $\omega$ represents the maximum degree of the vertices and size of maximum clique respectively.