T-COLORING ON FOLDED HYPERCUBES

Abstract

Given a graph $G=(V,E)$ and a set $T$ of non-negative integers containing $0$, a $T$-coloring of $G$ is an integer function $f$ of the vertices of $G$ such that $|f(u)-f(v)| \notin T$ whenever $uv \in E$. The edge-span of a $T$-coloring $f$ is the maximum value of $|f(u)-f(v)|$ over all edges $uv$, and the $T$-edge-span of a graph $G$ is the minimum value of the edge-span among all possible $T$-colorings of $G$. This paper discusses the $T$-edge span of the folded hypercube network of dimension $n$ for the $k$-multiple-of-$s$ set, $T=\{ 0$, $s$, $2s$, $\dots$, $ks\} \cup S$, where $s$ and $k \geq 1$ and $S \subseteq \{ s+1, s+2, \dots , ks-1 \}$.