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-edge- magic 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.
Highlights
We study undirected graphs without loops or multiple edges
The cycle will be called the rim of the wheel, and the edges connecting the hub to the vertices of the rim will be called the spokes. i.e., Wn = Cn + K1
We will use the term edge-magic plane graph for what was defined as edge-magic graph in [7], to differ it from other definitions of edge-magic graph
Summary
We study undirected graphs without loops or multiple edges. Given a graph G; V (G), E(G), v(G) and e(G) stands for the set of vertices, the set of edges, the order (number of vertices) and the size (number of edges) of G. Cn stand for the complete graph and the cycle of order n. For two graphs G and H we denote by G + H the graph obtained from the disjoint union G ∪· H by adding all edges between G and H. A wheel, Wn, is a graph of order n + 1 composed of a vertex, which will be called the hub, adjacent to all vertices of a cycle of order n. The cycle will be called the rim of the wheel, and the edges connecting the hub to the vertices of the rim will be called the spokes.
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