A Solution to the Edge-Balanced Index Set Problem for Complete Odd Bipartite Graphs

TL;DR

This paper provides a comprehensive solution to the edge-balanced index set problem specifically for all complete bipartite graphs with odd m and n, resolving a longstanding open problem in graph theory.

Contribution

It offers a general solution to the edge-balanced index set problem for all complete odd bipartite graphs, completing previous partial results.

Findings

Solved the edge-balanced index set problem for all complete odd bipartite graphs.

Extended previous work to cover all cases for odd bipartite graphs.

Concluded the problem for the case of complete odd bipartite graphs.

Abstract

In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets (EBI) 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 2011, Krop and Sikes expanded upon that work by finding EBI(K_{m,m-2a}) for odd m > 5 and 1 <= a <= (m-3)/4. In this paper, we provide a general solution to the edge-balanced index set problem for all complete odd bipartite graphs, thereby concluding the problem for this case.