On the edge-magic indices of (v, v+1)-graphs

Abstract

Let G be a (v, q)-graph. If the edges can be labeled 1,2, 3,...,q so that the vertex sums are constant (mod v), then G is said to be edge-magic. A necessary condition of edge-magicness is v | q(q +1). In [9], it was shown that for any graph G there is an integer k such that the k-fold graph G[k] is edge-magic. The least such integer k is called the edge-magic index of G. We characterize k for some (v, v + 1)-graphs. If v = pⁿ, where p is a prime, then any (pⁿ, pⁿ + 1)-graph has pⁿ - 1 as its edge-magic index. We also show here that there are infinitely many (v, v + 1)-graphs with edge-magic index less than v -1.