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 = |E(G)| and v = |V(G)|. A one-to-one map A from V ∪ E onto the integers {1, 2,..., v + e} is called vertex-magic total labeling if there is a constant k so that for every vertex λ(x) + σ λ (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 ωλ(x) of the vertex x under labeling λ. We require ωλ(x) = k for all x. The constant k is called the magic constant for λ. In this paper it is proved that the helm H_n 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 J_n,t+1 is vertex-magic for n = 3 and t = 1 but it has not this property for n ≥ 7t + 11 and t ≥ 1.