# On vertex-disjoint paths in regular graphs

## TL;DR

This paper proves that large regular graphs contain nearly spanning collections of vertex-disjoint paths, with bounds depending on the regularity ratio, and improves these bounds for bipartite graphs, establishing their optimality.

## Contribution

It establishes optimal bounds on the number of vertex-disjoint paths covering almost all vertices in regular and bipartite regular graphs.

## Key findings

- Regular graphs contain nearly spanning collections of vertex-disjoint paths.
- Bounds on the number of paths are tight and depend on the regularity ratio.
- Bipartite regular graphs have fewer such paths, with bounds proven to be optimal.

## Abstract

Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\lfloor 1/c \rfloor$ is best possible when $1/c\notin \mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\lfloor 1/(2c) \rfloor$, which is best possible.