Further results on super edge-magic deficiency of graphs

Abstract

Acharya and Hegde have introduced the notion of strongly k-indexable graphs: A (p, q)-graph G is said to be strongly k-indexable if its vertices can be assigned distinct integers 0, 1, 2, ..., p - 1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices can be arranged as an arithmetic progression k, k + 1, k + 2, ..., k + (q - 1). Such an assignment is called a strongly k-indexable labeling of G. Figueroa-Centeno et.al, have introduced the concept of super edge-magic deficiency of graphs: Super edge-magic deficiency of a graph G is the minimum number of isolated vertices added to G so that the resulting graph is super edge-magic. They conjectured that the super edge-magic deficiency of the complete bipartite graph K<inf>m,n</inf>, is (m -1)(n - 1) and proved it for the case m = 2. In this paper we prove that the conjecture is true for m = 3, 4 and 5, using the concept of strongly k-indexable labelings.