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

TL;DR

This paper establishes new upper bounds for $L(p,q)$-labeling numbers of various interval and circular-arc graph classes using greedy algorithms, improving previous bounds for specific graph types.

Contribution

It introduces simple greedy algorithms to derive upper bounds on $L(p,q)$-labeling numbers for multiple graph classes with interval representations, improving existing bounds.

Findings

Upper bounds for $L(p,q)$-labeling on interval and circular-arc graphs.

Improved bounds for $L(2,1)$-labeling of permutation graphs.

Upper bounds on coloring the squares of these graph classes.

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.