Spanning 3-Ended Trees in Almost Claw-Free Graphs

Xiaodong Chen,Meijin Xu,Yanjun Liu

doi:10.1155/2015/476182

Abstract

We prove that ifGis ak-connected (k≥2) almost claw-free graph of ordernandσk+3(G)≥n+2k-2, thenGcontains a spanning 3-ended tree, whereσk(G)=min⁡{∑v∈S‍deg⁡(v):Sis an independent set ofGwithS=k}.

Highlights

In this paper, only finite and simple graphs are considered
We prove that if G is a k-connected (k ≥ 2) almost claw-free graph of order n and σk+3(G) ≥ n + 2k − 2, G contains a spanning 3-ended tree, where σk(G) = min{∑V∈S deg(V) : S is an independent set of G with |S| = k}
We use γ(G) to denote the domination number of a graph G, where γ(G) = min{|S| : S is a dominating set of G}

Summary

Introduction

Only finite and simple graphs are considered. We refer to [1] for notation and terminology not defined here. Gave some degree sum conditions for K1,4-free graphs to contain spanning k-ended trees. If G is a connected claw-free graph of order n and σk+1(G) ≥ n − k (k ≥ 2), G contains spanning k-ended trees with the maximum degree at most 3. Chen et al [17] gave some degree sum conditions for k-connected K1,4-free graphs to contain spanning 3-ended trees. Inspired by Theorems 4 and 5, in this paper, we further explore sufficient conditions for k-connected almost clawfree graphs to contain spanning 3-ended trees which holds for claw-free graphs. If G is a k-connected almost claw-free graph of order n and σk+3(G) ≥ n + 2k − 2 (k ≥ 2), G contains spanning 3-ended trees. There are a lot of almost claw-free graphs containing K1,4 subgraphs, so to some extent Theorem 5 is a generalization of Theorem 4

Preliminaries

Proof of Theorem 5