Even Harmonious Labelings of Disconnected Graphs

Abstract

A graph $G$ with $q$ edges is called \textit{graceful} if there is an injection from the vertices of $G$ to the set $\{0,1,\ldots,q\}$ such that when each edge $xy$ is assigned the label $|f(x)-f(y)|$, the resulting edge labels are distinct. This notion as well as a number of other functions from a graph to a set of non-negative integers were studied as tools for decomposing the complete graph into isomorphic subgraphs. A graph $G$ with $q$ edges is said to be harmonious if there is an injection $f$ from the vertices of $G$ to the group of integers modulo $q$ such that when each edge $xy$ is assigned the label $f(x)+f(y)$ $($mod $q)$, the resulting edge labels are distinct. If $G$ is a tree, exactly one label may be used on two vertices. Over the years, many variations of these two concepts have been studied and hundreds of articles have been written on these topics. We study a variant of harmonious labeling. A function $f$ is said to be an even harmonious labeling of a graph $G$ with $q$ edges if $f$ is an injection from the vertices of $G$ to the integers from $0$ to $2q$ and the induced function $f^*$ from the edges of $G$ to ${0,2,\ldots,2(q-1)}$ defined by $f^*(xy)=f(x)+f(y) ($mod $2q)$ is bijective. Only a few papers have been written on even harmonious labeling. This paper focuses on finding even harmonious labelings for disjoint graphs. Among the families we investigate are: the disjoint union of cycles and stars, unions of cycles with paths, and unions of squares of paths.