On Mod(2) and Mod(3)-Edge-Magic Maximal Outerplanar Graphs

Abstract

Let $G$ be a $(p,q)$-graph. Suppose an edge labeling of $G$ given by $f:E(G)\to \{1,2,\dots ,q\}$ is a bijective function. For a vertex $v\in V(G)$, the induced vertex labeling of $G$ is a function $f^{\ast }(V)=\sum f(uv)$ for all $uv\in E(G)$. We say $f^{\ast }(V)$ is the vertex sum of $V$. If, for all $v\in V(G)$, the vertex sums are equal to a constant (mod $k$) where $k\ge 2$, then we say $G$ admits a Mod($k$)-edge-magic labeling, and $G$ is called a Mod($k$)-edge-magic graph. In this paper, we show that (i) all maximal outerplanar graphs (or MOPs) are Mod($2$)-EM, and (ii) many Mod($3$)-EM labelings of MOPs can be constructed (a) by adding new vertices to a MOP of smaller size, or (b) by taking the edge-gluing of two MOPs of smaller size, with a known Mod($3$)-EM labeling. These provide us with infinitely many Mod($3$)-EM MOPs. We conjecture that all MOPs are Mod($3$)-EM.