On Super Edge-Magic Graphs

Abstract

A \((p,q)\) graph \(G\) is edge-magic if there exists a bijective function \(f : V(G) \cup E(G) \to \{1,2,\ldots,p+q\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant, called the valence of \(f\), for any edge \(uv\) of \(G\). Moreover, \(G\) is said to be super edge-magic if \(f(V(G)) = \{1,2,\ldots,p\}\). Every super edge-magic \((p,q)\) graph is cordial, and it is harmonious and sequential whenever it is a tree or \(q \geq p\). In this paper, it is shown to be edge-antimagic as well. The super edge-magic properties of several classes of connected and disconnected graphs are studied. Furthermore, we prove that there can be arbitrarily large gaps among the possible valences for certain super edge-magic graphs. We also establish that the disjoint union of multiple copies of a super edge-magic linear forest is super edge-magic if the number of copies is odd.