# Bipartite partition-connected factors with small degrees

## TL;DR

This paper proves that highly connected graphs contain bipartite factors with small degrees and specific connectivity properties, extending understanding of graph decompositions and their degree constraints.

## Contribution

It introduces new results on bipartite $m$-partition-connected factors with bounded degrees in highly connected graphs.

## Key findings

- Existence of bipartite $m$-partition-connected factors with degree at most 75% of original degrees
- Bipartite $m$-partition-connected factors with maximum degree at most 3m+1 in tough graphs
- Presence of bipartite connected factors with degrees in {k, 2k, 3k, 4k} for large enough graphs

## Abstract

In this paper, we show that every $2m$-partition-connected graph $G$ has a bipartite $m$-partition-connected factor $H$ such that for each vertex $v$, $d_H(v)\le \lceil \frac{3}{4}d_G(v)\rceil$. A graph $H$ is said to be $m$-partition-connected, if it contains $m$ edge-disjoint spanning trees. As an application, we conclude that tough enough graphs with appropriate number of vertices have a bipartite $m$-partition-connected factor with maximum degree at most $3m+1$. Finally, we prove that tough enough graphs of order at least $3k$ admit a bipartite connected factor whose degrees lie in the set $\{k,2k,3k,4k\}$.