On vertex-magic total labeling of some wheel related graphs

Abstract

Let G be a graph with vertex set V = V (G) and edge set E = E(G) and let e = jE(G)j and v = jV (G)j. A one-to-one map ¸ from V [ E onto the integers f1; 2; : : : ; v + eg is called vertex-magic total labeling if there is a constant k so that for every vertex ¸(x) +X¸(xy) = k; where the sum is over all vertices y adjacent to x. Let us call the sum of labels at vertex x the weight !¸(x) of the vertex x under labeling ¸. We require !¸(x) = k for all x. The constant k is called the magic constant for ¸. In this paper it is proved that the helm Hn has no vertex-magic total labeling for any n ¸ 3. Also the generalized web WB(n; t) has a vertex-magic total labeling for n = 3 or n = 4 and t = 1 but it is not vertex-magic for n ¸ 17t + 12 and t ¸ 0. The generalized Jahangir graph Jn;t+1 is vertex-magic for n = 3 and t = 1 but it has not this property for n ¸ 7t + 11 and t ¸ 1.