eu WHEEL AS A VERTEX-EDGE-MAGIC PLANE GRAPH

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.


Introduction
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. K n , and C n 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, W n , 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.i.e., W n = C n + K 1 .

Magic Plane Graphs
Koh wei lih defined in [7] the notions of magic labeling of a plane graph.In this paper, 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.
Definition.Let G be a plane graph of size m.A bijection f : E(G) → {1, 2, ., m} is called edge-magic labeling of G if the sum of the edge labels surrounding each s-sided face of G is a constant.

Definition.
A plane graph G is called edge-magic plane graph if there exist an edge-magic labeling of G.
Definition.Let G be a plane graph such that all its bounded faces having the same size.G will be called k-edge-magic plane graph if there exist an edgemagic labeling of G, such that the sum of labels surrounding each face of G is k.
Notation.For a plane graph G, such that all its bounded faces having the same size, we denote by EM (G) the set of natural numbers, k, such that G as k-edge-magic labeling.
Two results have been shown in [2] regarding these concepts: Theorem 1.1.1For any odd natural number n ≥ 3, Theorem 1.1.2For any odd natural number n ≥ 3, Definition.Let G be a plane graph of order n and size m.A bijection f : V (G) ∪ E(G) → {1, 2, ., n + m} is called vertex-edge-magic labeling if the sum of the edge labels surrounding each s-sided face of G is a constant.

Definition.
A plane graph G is called vertex-edge-magic plane graph if there exist a vertex-edge-magic labeling of G.
Definition.Let G be a plane graph such that all its bounded faces having the same size.G will be called k-vertex-edge-magic plane graph if there exist a vertex-edge-magic labeling of G, such that the sum of labels surrounding each face of G is k.
Notation.For a plane graph G, such that all its bounded faces having the same size, we denote by V EM (G) the set of natural numbers, k, such that G as k-vertex-edge-magic labeling.
Ko wei lih shows in [7] that for all n ≥ 3, W n has a consecutive vertex labeling if and only if n ≡ 2 mod 4. In addition he shows that for all n ≥ 3, W n has a consecutive edge labeling if and only if n ≡ 2 mod 4. From these last two results of K.W. Lih it is easy to deduce that for all n ≥ 3, n ≡ 2 mod 4, W n has an edge-vertex magic labeling.
On this paper we will find min(V EM (W n )) and max(V EM ((W n )) for all odd natural number n.

Labeling of Wheels
Let (a 1 , . . ., a n ) be the labeling of the spokes, (b 1 , . . ., b n ) the labeling of the rim edges, (c 1 , . . ., c n ) the labeling of the rim vertices and c n+1 the labeling of the hub of W n , such that the sum of the labels on each face of the wheel is k.Since each spoke and each rim vertex belongs to two faces, each rim edge belongs to only one face and the hub belongs to n faces, we conclude that: Hence, it is easy to derive the following inequalities Since, Thus, In the case of odd n, we will show that k attain these bounds.
Theorem 2.1.For any odd natural number n ≥ 3, Proof.Let m be the natural number, such that n = 2m+1.From inequality (2) it is sufficient to point out a 13m + 15 vertex-edge-magic labeling of W 2m+1 for all natural m.Such a labeling of W 2m+1 can be described as followed.We label the hub vertex by 1 and the spokes edges by 2, 3, 4, .., 2m + 1, 2m + 2 clockwise.We label the rim vertices 2m + 3, 2m + 4, .., 4m + 3 counter clockwise skipping one edge every time starting by labeling 2m + 3 the rim vertex, which is contained in the spoke labeled 2m + 1.The rim edges we label 4m + 4, 4m + 5, . . ., 6n + 4) counter clockwise, starting by labeling by 4m + 4 the rim edge containing the vertices, labeled by 4m + 3 and 3m + 3.Such a labeling is demonstrated by Figure 1.Notice that the sum of labels on the triangle which his vertices labeled by 1, 4m + 3, 3m + 3 is indeed 13m + 15 and that is also the sum of labels on the adjacent triangle counter clockwise.It is easy to see that from there on, Moving from a triangle to the adjacent triangle counter clockwise, the sum of the labels stays constant.Theorem 2.2.For any odd natural number n ≥ 3, Proof.Let m be the natural number, such that n = 2m+1.From inequality (2) it is sufficient to point out a 13m + 15 vertex-edge-magic labeling of W 2m+1 for all natural m.Such a labeling of W 2m+1 can be described as followed.We label the hub vertex by 6m + 4 and the spokes edges by 4m + 3, 4m + 4, 4m + 5, .., 6m + 3 clockwise.We label the rim vertices 2m + 2, 2m + 3, .., 4m + 2 counter clockwise skipping one vertex every time starting by labeling the rim vertex, which is contained in the spoke labeled 6m + 2. The rim edges we label 1, 2, . . ., 2m + 1 counter clockwise, starting with the rim edge containing the vertices, labeled by 4m + 2 and 3m + 2. Such a labeling is demonstrated by Figure 3. Notice that the sum of the labels on the triangle which his vertices labeled by 6m + 4, 4m + 2, 3m + 2 is indeed 13m + 15 and that is also the sum of labels on the adjacent triangle counter clockwise.It is easy to see that from there on, Moving from a triangle to the adjacent triangle counter clockwise, the sum of the labels stays constant.
Figure 1.describes an edge-vertex magic labeling of W 2m+1 with a minimum magic number.Figure 2. demonstrate such a labeling on W 7 .
Figure 3. describes an edge-vertex magic labeling of W 2m+1 with a maximum magic number.Figure 4. demonstrate such a labeling on W 7 .

Discussion
We saw that for any odd natural number n ≥ 3, The question is whether these formulas are valid also in the case of even numbers.Moreover, for n ≡ 2 mod 4 it is needed first to prove that for any such n there exist an edge-vertex magic labeling of W n .the following figures shows that these minimum and maximum values are valid at least for n = 4.