Regular spanning subgraphs of bipartite graphs of high minimum degree

TL;DR

This paper proves that bipartite graphs with high minimum degree contain large regular spanning subgraphs, establishing near-tight bounds related to the minimum degree and graph size.

Contribution

It introduces a new bound on the size of regular spanning subgraphs in bipartite graphs with high minimum degree, extending previous results.

Findings

Bipartite graphs with minimum degree greater than half their vertices contain large regular spanning subgraphs.

The bound on the regular subgraph size is nearly optimal.

Provides a mathematical relationship between minimum degree and regular subgraph size.

Abstract

Let G be a simple balanced bipartite graph on $2n$ vertices, $\delta = \delta(G)/n$, and $\rho={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta > 1/2$ then it has a $\lfloor \rho n \rfloor$-regular spanning subgraph. The statement is nearly tight.