Spanning trees with at most $5$ leaves and branch vertices in total of $K_{1,5}$-free graphs

TL;DR

This paper proves that certain large, connected, $K_{1,5}$-free graphs always contain a spanning tree with at most 5 leaves and branch vertices, under a specific degree sum condition, which is proven to be optimal.

Contribution

It establishes a new degree sum condition ensuring the existence of a spanning tree with limited leaves and branch vertices in $K_{1,5}$-free graphs, and proves the condition's optimality.

Findings

Every $n$-vertex connected $K_{1,5}$-free graph with $\sigma_4(G) extgreater=n-1$ has a spanning tree with at most 5 leaves and branch vertices.

The degree sum condition $\sigma_4(G) extgreater=n-1$ is sharp and cannot be lowered.

The result extends understanding of spanning trees in restricted graph classes.

Abstract

In this paper, we prove that every $n$-vertex connected $K_{1,5}$-free graph $G$ with $\sigma_4(G)\geq n-1$ contains a spanning tree with at most $5$ leaves and branch vertices in total. Moreover, the degree sum condition "$\sigma_4(G)\geq n-1$" is best possible.