On kk-edge-magic labelings of maximal outerplanar graphs

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