Spanning weakly even trees of graphs

Abstract

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\text{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\text{weakly even}$ if every leaf of $T$ that has maximum degree in $G$ belongs to the same part of the bipartition of $T$. We confirm two recent conjectures of Jackson and Yoshimoto by showing that every connected graph that is not a regular bipartite graph has a spanning weakly even tree.

6 pages. This article represents a merger of arXiv:2409.15522v1 and arxiv:2408.07056