Spanning trees with at most 2 branch vertices in claw - free graphs

TL;DR

This paper proves that certain degree sum conditions in connected claw-free graphs guarantee the existence of a spanning tree with at most two branch vertices, extending understanding of graph structures under specific constraints.

Contribution

It establishes new sufficient conditions involving degree sums for the existence of spanning trees with limited branch vertices in claw-free graphs.

Findings

Graphs with specified degree sum conditions have spanning trees with at most two branch vertices.

The results apply to connected claw-free graphs under particular degree sum thresholds.

The paper extends known conditions for spanning trees with constrained branch vertices.

Abstract

In this article, we will prove that if $G$ is a connected claw-free graph and either $\sigma_6(G)\geq |G|-5$ or $\sigma_7(G)\geq |G|-2$, here $\sigma_k(G)$ is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a spanning tree with at most two branch vertices.