On the (Consecutively) Super Edge-Magic Deficiency of Subdivision of Double Stars

Abstract

Let G be a finite, simple, and undirected graph with vertex set VG and edge set EG. A super edge-magic labeling of G is a bijection f:VG∪EG⟶1,2,…,VG+EG such that fVG=1,2,…,VG and fu+fuv+fv is a constant for every edge uv∈EG. The super edge-magic labeling f of G is called consecutively super edge-magic if G is a bipartite graph with partite sets A and B such that fA=1,2,…,A and fB=A+1,A+2,…,VG. A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency of G, denoted by μsG, is either the minimum nonnegative integer n such that G∪nK1 is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a graph G is defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.