On Super Edge-Magic Graphs

Abstract

A $(p,q)$ graph $G$ is edge-magic if there exists a bijective function $f:V(G)\cup E(G)\to \{1,2,\dots ,p+q\}$ 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 $f(V(G))=\{1,2,\dots ,p\}$. Every super edge-magic $(p,q)$ graph is cordial, and it is harmonious and sequential whenever it is a tree or $q\ge p$. 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.