A neighborhood union condition for the existence of a spanning tree without samll degree vertices

TL;DR

This work addresses the existence of $[2,k]$-STs in connected graphs under a neighborhood-union condition. It introduces i-semi-$[2,k]$-T and i-quasi-$[2,k]$-T to structure partial solutions and proves an extension principle that combines disjoint pieces into a spanning $[2,k]$-ST. The main contribution is a large-n result: for $n\ge n_1(k)$, if $\delta(G)\ge 2k$ and $NC(G)\ge (n-2)/2$, then $G$ has a $[2,k]$-ST, generalizing prior degree-sum conditions and providing a sharp extremal example to illustrate tightness. The analysis hinges on a careful case split on the connectivity of $G[W]$, leveraging neighborhood structure and existing $[2,k]$-ST results to assemble a global spanning tree without vertices of degrees in $[2,k]$.

Abstract

For an integer k\ge2, a [2,k]-ST of a connected graph G is a spanning tree of G in which there are no vertices of degree between 2 and k. A [2,k]-ST is a natural extension of a homeomorphically irreducible spanning tree (HIST), which is a spanning tree without vertices of degree 2. In this paper, we give a neighborhood union condition for the existence of a [2,k]-ST in G. We generalize a known degree sum condition that guarantees the existence of a [2,k]-ST in G.

Figures (4)

• Figure 1: $H$ is obtained from vertex-disjoint graphs $K_{2k-1}$, $K_{\left \lceil \frac{n+1}{2}\right \rceil-k}$ and $K_{\left \lfloor \frac{n+1}{2}\right \rfloor-k}$ by adding edges $uv$ and $u_iw_i$ for $i=1,2,\cdots,2k-2$
• Figure 2: An $i$-semi-$[2,k]$-T $T$ with $d_T(v_j)=k$ for all $j=1,2,\cdots,i$
• Figure 3: A 1-semi-$[2,k]$-T $T$ of $G$ with vertex set $N[u]\cup S$
• Figure 4: Two subtrees of $G$ when $k=3$

Theorems & Definitions (17)

• Theorem 1.1: ABHT90
• Theorem 1.2: ItTs22
• Theorem 1.3: LiDo24
• Theorem 1.4: FuST24
• Theorem 1.5: FuTsarxiv
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4