Super edge-magic labeling of graphs: deficiency and maximality - Anak Agung Gede Ngurah; Rinovia Simanjuntak (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to 1, 2, 3, ..., p+q such that f(x) + f(xy) + f(y) is a constant for every edge xy E(G) and f(V(G)) = 1, 2, 3, ..., p. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or + if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.