On Edge-Graceful and Edge-Magic Maximal Outerplanar Graphs

Abstract

Let $G$ be a $(p,q)$-graph in which the edges are labeled $1,2,3,\dots ,q$. The vertex sum for a vertex $v$ is the sum of the labels of the incident edges at $v$. If $G$ can be labeled so that the vertex sums are distinct, mod $p$, then $G$ is said to be edge-graceful. If the edges of $G$ can be labeled $1,2,3,\dots ,q$ so that the vertex sums are constant, mod $p$, then $G$ is said to be edge-magic. It is conjectured by Lee [9] that any connected simple $(p,q)$-graph with $q(q+1)\equiv p(p-1)/2(\mathrm{mod}p)$ vertices is edge-graceful. We show that the conjecture is true for maximal outerplanar graphs. We also completely determine the edge-magic maximal outerplanar graphs.