ON CIRCULAR-$L(2,1)$-EDGE-LABELING OF GRAPHS

Abstract

Let $m$, $j$ and $k$ be positive integers with $j\ge k$. An $m$-circular-$L(j,k)$-edge-labeling of a graph $G$ is an assignment $f$ from $\{0,1,\dots,m-1\}$ to the edges of $G$ such that, for any two edges $e_1$ and $e_2$, $|f(e_1)-f(e_2)|_m\geq j$ if $e_1$ and $e_2$ are adjacent, and $|f(e_1)-f(e_2)|_m\geq k$ if $e_1$ and $e_2$ are at distance $2$, where $|a|_m=\min\{a,m-a\}$. The minimum $m$ such that $G$ has an $m$-circular-$L(j,k)$-edge-labeling is defined as the circular-$L(j,k)$-edge-labeling number of $G$, denoted by $\sigma_{j,k}'(G)$. This paper determines the circular-$L(2,1)$-edge-labeling numbers of the infinite $\Delta$-regular tree for $\Delta\ge 2$ and the $n$-dimensional cube for $n\in \{2,3,4,5\}$.