On trees with a maximum proper partial 0-1 coloring containing a maximum matching

TL;DR

This paper proves that in certain trees with even-distance endpoints, there exists a maximum proper partial 0-1 coloring where edges colored 0 constitute a maximum matching, linking coloring properties to maximum matchings.

Contribution

It establishes a specific coloring property in trees with even-distance endpoints, connecting maximum proper partial 0-1 colorings to maximum matchings.

Findings

Existence of a maximum proper partial 0-1 coloring with edges colored 0 forming a maximum matching in such trees.

The property holds for trees where the distance between any two endpoints is even.

Abstract

I prove that in a tree in which the distance between any two endpoints is even, there is a maximum proper partial 0-1 coloring such that the edges colored by 0 form a maximum matching.