Distance Two Surjective Labelling of Paths and Interval Graphs

Abstract

Graph labelling problem has been broadly studied for a long period for its applications, especially in frequency assignment in (mobile) communication system, $X$ -ray crystallography, circuit design, etc. Nowadays, surjective $L\left(\mathrm{2,1}\right)$ -labelling is a well-studied problem. Motivated from the $L\left(\mathrm{2,1}\right)$ -labelling problem and the importance of surjective $L\left(\mathrm{2,1}\right)$ -labelling problem, we consider surjective $L\left(\mathrm{2,1}\right)$ -labelling ( $\text{SL}21$ -labelling) problems for paths and interval graphs. For any graph $G=\left(V,E\right)$ , an $\text{SL}21$ -labelling is a mapping $\phi :V⟶\left\{\mathrm{1,2},\dots ,n\right\}$ so that, for every pair of nodes $u$ and $v$ , if $d\left(u,v\right)=1$ , then $\left|\phi \left(u\right)-\phi \left(v\right)\right|\ge 2$ ; and if $d\left(u,v\right)=2$ , then $\left|\phi \left(u\right)-\phi \left(v\right)\right|\ge 1$ , and every label $\mathrm{1,2},\dots ,n$ is used exactly once, where $d\left(u,v\right)$ represents the distance between the nodes $u$ and $v$ , and $n$ is the number of nodes of graph $G$ . In the present article, it is proved that any path $P_n$ can be surjectively $L\left(\mathrm{2,1}\right)$ -labelled if $n\ge 4$ , and it is also proved that any interval graph $\left(\text{IG}\right)$ $G$ having $n$ nodes and degree $\mathrm{\Delta }>2$ can be surjectively $L\left(\mathrm{2,1}\right)$ -labelled if $n=3\mathrm{\Delta }-1$ . Also, we have designed two efficient algorithms for surjective $L\left(\mathrm{2,1}\right)$ -labelling of paths and interval graphs. The results regarding both paths and interval graphs are the first result for surjective $L\left(\mathrm{2,1}\right)$ -labelling.