$L(p,q)$-Labeling of Graphs with Interval Representations

Abstract

We provide upper bounds on the $L(p,q)$-labeling number of graphs which have interval (or circular-arc) representations via simple greedy algorithms. We prove that there exists an $L(p,q)$-labeling with span at most $\max \{2(p+q-1)\Delta -4q+2,(2p-1)\mu +(2q-1)\Delta -2q+1\}$ for interval $k$-graphs, $\max \{p,q\}\Delta$ for interval graphs, $3\max \{p,q\}\Delta +p$ for circular-arc graphs, $2(p+q-1)\Delta -2q+1$ for permutation graphs and $(2p-1)\Delta +(2q-1)(\mu -1)$ for cointerval graphs. In particular, these improve existing bounds on $L(p,q)$-labeling of interval graphs and $L(2,1)$-labeling of permutation graphs. Furthermore, we provide upper bounds on the coloring of the squares of aforementioned classes.

21 pages, 8 figures. This is the final version that will appear in Discussiones Mathematicae Graph Theory