Spanning k-ended trees of 3-regular connected graphs

TL;DR

This paper proves that large 3-regular connected graphs always contain spanning subtrees with a limited number of leaves, specifically at most [(2n+4)/9], and proposes a conjecture on spanning k-ended trees.

Contribution

It introduces a new bound for the existence of spanning k-ended trees in 3-regular connected graphs and presents a related conjecture.

Findings

Every 3-regular connected graph with n > 8 vertices has a spanning subtree with at most [(2n+4)/9] leaves.

The paper proposes a conjecture about the existence of spanning k-ended trees in such graphs.

Abstract

A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most [(2n+4)/9]-ended tree. At the end we give a conjecture about spanning k-ended trees on 3-regular connected graphs.