On the edge-balanced index set of complete bipartite graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $f$ be a $0-1$ labeling of $E(G)$ so that the absolute difference in the number of edges labeled $1$ and $0$ is no more than one. Call such a labeling $f$ \emph{edge-friendly}. The \emph{edge-balanced index set} of the graph $G$, $EBI(G)$, is defined as the absolute difference between the number of vertices incident to more edges labeled $1$ and the number of vertices incident to more edges labeled $0$ over all edge-friendly labelings $f$. In 2009, Lee, Kong, and Wang \cite{LeeKongWang} found the $EBI(K_{l,n})$ for $l=1,2,3,4,5$ as well as $l=n$. We continue the investigation of the $EBI$ of complete bipartite graphs of other orders.