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 \(0,1,2,\ldots,p-1\) 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 \(k,k+1,k+2,\ldots,k+(q-1)\). 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 \(K_{m,n}\) is \((m-1)(n-1)\) and proved it for the case \(m=2\). In this paper, we prove that the conjecture is true for \(m=3,4,5\), using the concept of strongly \(k\)-indexable labelings \(^1\).