$L(h,k)$ labelings of $K_n-M$ and $K_n-P_m$ for all values of $h$ and $k\ast$

Abstract

An $L(h,k)$ labeling of a graph $G$ is an integer labeling of the vertices where the labels of adjacent vertices differ by at least $h$, and the labels of vertices that are at distance two from each other differ by at least $k$. The span of an $L(h,k)$ labeling $f$ on a graph $G$ is the largest label minus the smallest label under $f$. The $L(h,k)$ span of a graph $G$, denoted ${\lambda }_{h,k}(G)$, is the minimum span of all $L(h,k)$ labelings of $G$.