A short note on spanning even trees

TL;DR

This paper proves that every odd-regular nonbipartite connected graph contains a spanning even tree, confirming a conjecture for the remaining unresolved case and advancing understanding of graph structures.

Contribution

It establishes the conjecture for odd-regular graphs, resolving the only remaining case and extending previous results.

Findings

Confirmed the conjecture for odd-regular graphs

Resolved the last open case of the conjecture

Extended the understanding of spanning even trees

Abstract

We call a tree $T$ is \emph{even} if every pair of its leaves is joined by a path of even length. Jackson and Yoshimoto~[J. Graph Theory, 2024] conjectured that every $r$-regular nonbipartite connected graph $G$ has a spanning even tree. They verified this conjecture for the case when $G$ has a $2$-factor. In this paper, we prove that the conjecture holds when $r$ is odd, thereby resolving the only remaining unsolved case for this conjecture.