On k-edge-magic labelings of maximal outerplanar graphs

Abstract

Let G be a graph with vertex set V and edge set E such that |V|=p and |E|=q. We denote this graph by (p,q)-graph. For integers k≥0, define a one-to-one map f from E to {k,k+1,…,k+q−1} and define the vertex sum for a vertex v as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (modp), then G is said to be k-edge magic (k-EM). In this paper, we show that a maximal outerplanar graph of orders p = 4, 5, 7 are k-EM if and only if k≡2(modp) and obtain all maximal outerplanar graphs that are k-EM for k = 3, 4. Finally we characterize all (p,p−h)-graphs that are k-EM for h≥0. We conjecture that a maximal outerplanar graph of prime order p is k-EM if and only if k≡2(modp).

Keywords

• Edge-magic
• Maximal outerplanar graph
• Labeling