Spanning trees of $K_{1,4}$-free graphs with a bounded number of leaves and branch vertices

TL;DR

This paper investigates the properties of spanning trees with limited leaves and branch vertices in $K_{1,4}$-free graphs, providing new bounds and improvements over previous results.

Contribution

It introduces new bounds on spanning trees with few leaves and branch vertices specifically for $K_{1,4}$-free graphs, advancing understanding in this graph class.

Findings

Established bounds on spanning trees with limited leaves and branch vertices.

Improved previous results for $K_{1,4}$-free graphs.

Enhanced understanding of spanning tree structures in restricted graph classes.

Abstract

Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. A graph is said to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_ {1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.