On the Edge-Balanced Index Sets of Complete Even Bipartite Graphs

TL;DR

This paper determines the edge-balanced index sets for complete bipartite graphs with both parts of even size, extending previous work that focused on odd-sized parts.

Contribution

It provides the first complete characterization of edge-balanced index sets for complete bipartite graphs with both parts even, filling a gap in existing research.

Findings

Derived explicit formulas for $EBI(K_{m,n})$ with even $m,n$

Extended the classification of $EBI$ to new graph cases

Filled a gap in the literature for even bipartite graphs

Abstract

In 2009, Kong, Wang, and Lee introduced the problem of finding the edge-balanced index sets ($EBI$) of complete bipartite graphs $K_{m,n}$, where they examined the cases $n=1$, $2$, $3$, $4$, $5$ and the case $m=n$. Since then the problem of finding $EBI(K_{m,n})$, where $m \geq n$, has been completely resolved for the $m,n=$ odd, odd and odd, even cases. In this paper we find the edge-balanced index sets for complete bipartite graphs where both parts have even cardinality.