Mathematical Problems in Engineering

Volume 2019, Article ID 1348348, 11 pages

https://doi.org/10.1155/2019/1348348

## Spanning Trees with At Most 6 Leaves in -Free Graphs

^{1}Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China^{2}Collage of Sciences, Henan University of Engineering, Zhengzhou 451191, China

Correspondence should be addressed to Pei Sun; moc.anis@kkiepnus

Received 10 February 2019; Revised 13 July 2019; Accepted 7 September 2019; Published 3 October 2019

Academic Editor: Jean Jacques Loiseau

Copyright © 2019 Pei Sun and Kai Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A graph *G* is called -free if *G* contains no as an induced subgraph. A tree with at most *m* leaves is called an *m*-ended tree. Let be the minimum degree sum of *k* independent vertices in *G*. In this paper, it is shown that every connected -free graph *G* contains a spanning 6-ended tree if .

#### 1. Introduction

We will begin with some basic definitions, notations, and terminologies used by Bondy and Murty [1]. The *graph* generally means a simple undirected graph *G* with a vertex set and an edge set . We write for the order of *G*. For any , let be the set of vertices adjacent to in *G* and . For a nonempty subset , we write

For an integer *i*, we define and . Let *T* be a tree. A vertex of degree one in *T* is called a leaf of *T*, and a vertex of degree at least three is called a branch vertex. If are vertices of *T*, the path in *T* connecting *u* and is unique and is denoted by . Define , and . The distance between *u* and in *T* is denoted by .

A subset is called an independent set of *G* if no two vertices of *A* are adjacent in *G*. The independence number of *G*, , is the order of the maximum independent set of *G*. For an integer *k*, we define

For an integer , a graph *G* is called -free if *G* contains no as an induced subgraph. A tree with at most *m* leaves is called an *m*-ended tree. If *T* is an *m*-ended tree in *G* such that is as large as possible, we call *T* the maximum *m*-ended tree of *G*. There are several well-known conditions ensuring that a graph *G* contains a spanning *m*-ended tree (see the survey paper [2]). In 1979, Win [3] obtained a sufficient condition related to an independent number for *k*-connected graphs, which confirms the conjecture of Las Vergnas [4].

Theorem 1 (Win [3]). *Let G be a k-connected graph, and let . If , then G has a spanning m-ended tree.*

There are many results on the degree sum conditions for a graph to contain a spanning tree with a bounded number of leaves or branch vertices. Among them are the following theorems.

Theorem 2 (Broersma and Tuinstra [5]). *Let G be a connected graph with n vertices, and let . If , then G has an m-ended tree.*

Theorem 3 (Gargano et al. [6]). *Let k be a nonnegative integer, and let G be a connected claw-free graph. If , then G has a spanning tree with at most k branch vertices.*

Theorem 4 (Kano et al. [7]). *Let k be a nonnegative integer, and let G be a connected claw-free graph. If , then G has a spanning tree with at most leaves.*

Theorem 5 (Kyaw [8]). *Every connected -free graph with contains a spanning tree with at most 3 leaves.*

Theorem 6 (Kyaw [9]). * Let G be a connected -free graph.*(1)

*If , then*(2)

*G*has a Hamiltonian path.*If for an integer , then*

*G*has a spanning tree with at most*k*leaves.*Theorem 7 (Chen et al. [10]). Let G be a connected -free graph with n vertices. If , then G contains a spanning 4-ended tree.*

*In this paper, we further consider -free graphs and generalize Theorem 7 as follows.*

*Theorem 8. Let G be a connected -free graph. If , then G contains a spanning 6-ended tree.*

*Before proving Theorem 8, we first give an example to show that the condition in Theorem 8 is sharp.*

*Let m be a positive integer, and let G be the graph obtained from a triangle and 7 copies of by joining x to each vertex of , y to each vertex of , and z to each vertex of (see Figure 1). It is easy to see that , but each spanning tree of G has at least 7 leaves. Therefore, the condition in Theorem 8 is sharp.*