Wheel as a Vertex-Edge-Magic Plane Graph

Abstract

An image of a plane graph, G = (V, E) of order n and size m, is said to be a vertex-edge-magic plane graph if there is a bijection f : V ∪ E → {1, 2, .., n + m} such that for all s − side faces of G, except the infinite face, the sum of the labels of its vertices and edges is a constant k(s). Such a bijection will be called a vertex-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 a vertex-edgemagic labeling of G, in which k is the sum of the vertex and edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order.