On $k$-Edge-Magic Cubic Graphs

Abstract

Let $G$ be a $(p,q)$-graph in which the edges are labeled $k,k+1,\dots ,k+q-1$, where $k\ge 0$. The vertex sum for a vertex $v$ is the sum of the labels of the incident edges at $v$. If the vertex sums are constant, modulo $p$, then $G$ is said to be $k$-edge-magic. In this paper, we investigate some classes of cubic graphs which are $k$-edge-magic. We also provide a counterexample to a conjecture that any cubic graph of order $p\equiv 2(\mathrm{mod}4)$ is $k$-edge-magic for all $k$.