On the Edge-balanced Index Sets of Complete Bipartite Graphs

TL;DR

This paper investigates the edge-balanced index sets of complete bipartite graphs, extending previous work by analyzing cases beyond those already studied, and provides new insights into their structure.

Contribution

It introduces new results on the edge-balanced index sets of complete bipartite graphs of various orders, expanding the known cases and deepening understanding.

Findings

Determined the edge-balanced index sets for new classes of complete bipartite graphs.

Extended previous results to broader graph orders.

Provided a comprehensive analysis of edge-friendly labelings.

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.