On $k$-ended spanning and dominating trees

TL;DR

This paper improves existing conditions for the existence of spanning trees with few leaves in graphs and introduces analogous results for dominating $k$-ended trees based on the size of the largest such trees.

Contribution

It enhances previous theorems on spanning $k$-ended trees and extends results to dominating $k$-ended trees using the parameter $t_k$.

Findings

Improved bounds for the existence of spanning $k$-ended trees.

Established conditions for dominating $k$-ended trees.

Connected graphs with certain degree sum conditions have large $k$-ended trees.

Abstract

A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that if $k$ is a positive integer and $G$ is a connected graph of order $n$ with $d(x)+d(y)\ge n-k+1$ for each pair of nonadjacent vertices $x,y$, then $G$ has a spanning $k$-ended tree. In this paper, we improve this result in two ways, and an analogous result is proved for dominating $k$-ended trees based on the generalized parameter $t_k$ - the order of a largest $k$-ended tree. In particular, $t_1$ is the circumference (the length of a longest cycle), and $t_2$ is the order of a longest path.