Spanning trees in a Claw-free graph whose stems have at most $k$ branch vertices

TL;DR

This paper establishes optimal conditions under which a connected claw-free graph contains a spanning tree with a stem having at most k branch vertices, and explores related structures like spiders.

Contribution

It provides two best-possible sufficient conditions for the existence of such spanning trees in claw-free graphs, extending understanding of graph structures.

Findings

Two optimal sufficient conditions for spanning trees with bounded stem branch vertices.

Conditions under which a spanning tree's stem is a spider.

Extension of results to specific graph classes like claw-free graphs.

Abstract

Let $T$ be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this paper, we give two sufficient conditions for a connected claw-free graph to have a spanning tree whose stem has a bounded number of branch vertices, and those conditions are best possible. As corollaries of main results we also give some conditions to show that a connected claw-free graph has a spanning tree whose stem is a spider.