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

TL;DR

This paper determines the edge-balanced index sets for complete bipartite graphs where the larger part has an odd number of vertices and the smaller part has an even number, completing a long-standing classification.

Contribution

It extends previous work by solving the edge-balanced index set problem for bipartite graphs with an odd larger part and an even smaller part.

Findings

Complete characterization for the specified bipartite graphs.

Resolution of the problem for the case with odd larger part and even smaller part.

Fills a gap in the classification of edge-balanced index sets.

Abstract

In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets of complete bipartite graphs $K_{m,n}$ by solving the cases where $n=1$, $2$, $3$, $4$, and $5$, and also the case where $m=n$. In an article soon to be published, Krop, Minion, Patel, and Raridan concluded the edge-balanced index set problem for complete bipartite graphs with both parts of odd cardinality. In this paper, we conclude the problem for complete bipartite graphs where the larger part is of odd cardinality and the smaller is of even cardinality.