Wheel as an Edge-Magic Graph

Abstract

An image of a plane graph, G = (V, E) of order n and size m, is said to be an edge-magic plane graph if there is a bijection f : E → {1, 2, .., m} such that for all s−side faces of G, except the infinite face, the sum of the labels of its edges is a constant k(s). Such a bijection will be called an edge-magic plane labeling of G. In case that all the finite sides of a graph G having the same size we will be interested in determining the minimum and the maximum number, k, such that there exists an edge-magic plane labeling of G, in which k is the sum of the edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order. Furthermore we conjecture that the same formula is valid for the odd case.