Spanning weakly even trees of graphs

TL;DR

This paper proves that every connected non-regular bipartite graph contains a spanning weakly even tree, confirming two recent conjectures in graph theory.

Contribution

It establishes the existence of spanning weakly even trees in a broad class of graphs, resolving two conjectures by Jackson and Yoshimoto.

Findings

Every connected non-regular bipartite graph has a spanning weakly even tree.

The result confirms two recent conjectures in graph theory.

The paper advances understanding of bipartite graph structures.

Abstract

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ 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.