Edge-magic Indices of Multigraphs Containing one more Vertices than Edges

Abstract

A graph G = (V; E) with p vertices and q edges is called edge-magic if there is a bijection f : E ! f1; 2; : : : ; qg such that the induced mapping f + : V ! Z p is a constant mapping, where f + (u) = X v2N(u) f(uv). A necessary condition of edge-magicness is pjq(q + 1). The edgemagic index of a graph G is the least positive integer k such that the k-fold of G is edge-magic. In this paper, we prove that for any multigraph G with n+ 1 vertices, n edges having no loops and no isolated vertices, the k-fold of G is edge-magic if n and k satisfy the necessary condition of edge-magicness and n is even. For n is odd we also have some results on full m-ary trees and spider graphs. Some counterexamples of the edge-magic indices of spider graphs conjecture are given