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 \( f: E(G) \to \{1, 2, \ldots, q\} \) is a bijective function. For a vertex \( v \in V(G) \), the induced vertex labeling of \( G \) is a function \( f^*(V) = \sum f(uv) \) for all \( uv \in E(G) \). We say \( f^*(V) \) is the vertex sum of \( V \). If, for all \( v \in V(G) \), the vertex sums are equal to a constant (mod \( k \)) where \( k \geq 2 \), 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.