Spanning trees of claw-free graphs with few leaves and branch vertices

TL;DR

The paper proves that a connected claw-free graph $G$ with $\sigma_{m+1}(G)\ge |G|-n+m-1$ and $m \le \lceil \dfrac{2n}{3} \rceil$ has a spanning tree with at most $n$ leaves and branch vertices. The proof uses a contradiction argument, selecting a spanning tree with minimal $|B(T)|$ and performing a case split depending on whether $|B_3(T)|=0$ or $|B_3(T)|>0$, together with oblique-neighbor and pseudoindependent-set techniques to control $\sigma_{m+1}(G)$. The authors also derive corollaries for $\sigma_{2}(G)$ and show improvements for claw-free and $K_{1,r}$-free graphs, connecting to prior results by Gargano et al., Hanh, Ha, Trang and others. Overall, the work unifies several conditions for spanning trees with few leaves/branch vertices in claw-free graphs and extends known bounds.

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 claw-free if it does not contain $K_{1,3}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of claw-free graphs. Applying the main results, we also give some improvements of previous results on the spanning trees with few branch vertices for the case of claw-free graphs.

Theorems & Definitions (26)

• Theorem 1.1: Gargano et al. GHHSV
• Theorem 1.2: Gould and Shull GS
• Theorem 1.3: Nikoghosyan N, Saito and Sano SS
• Theorem 1.4: Maezawa et al. MMM
• Theorem 1.5: Hanh Hanh1
• Theorem 1.6: Ha H2
• Theorem 1.7: Ha and Trang HT
• Theorem 1.8: Diep et al. DHT
• Theorem 1.9
• Corollary 1.10