$L(p,q)$-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.