A note on spanning trees with a specified degree sequence

TL;DR

This paper presents a new Ore-Type condition that guarantees the existence of a spanning tree with a given degree sequence in a graph, advancing understanding of graph structures and spanning trees.

Contribution

It introduces a novel Ore-Type condition that ensures a graph contains a spanning tree with a specified degree sequence, providing a new criterion for graph structure analysis.

Findings

Established a sufficient Ore-Type condition for spanning trees with a given degree sequence

Extended classical results to more general degree sequence constraints

Provided theoretical insights into graph spanning tree configurations

Abstract

We give an Ore-Type condition sufficient for a graph G to have a spanning tree with a specified degree sequence.

Equations

$$d_{G_k}(u)+d_{G_k}(v)=2((k-1)+(2k+2))=6k+2=\frac{3n-2}{2}$$

$$A_u=\{y^{-}\in V(T_u):uy\in E(G)\},B_u=\{x\in V(T_u):vx\in E(G)\},$$

$$A_v=\{y^{-}\in V(T_v):vy\in E(G)\}\text{ and }{B}_v=\{x\in V(T_v):ux\in E(G)\}.$$

$$|A_u|+|B_u|=|A_u\cup B_u|\le n_u\text{ and }|A_v|+|B_v|=|A_v\cup B_v|\le n_v,$$

$$|B_u|=d_u(v)\text{ and }|B_v|=d_v(u).$$

$$|A_u|\ge \frac{d_u(u)}{2}\text{ and }|A_v|\ge \frac{d_v(v)}{2}$$

$$\frac{d_u(u)}{2}+d_u(v)\le |A_u|+|B_u|\le n_u\text{ and }\frac{d_v(v)}{2}+d_v(u)\le |A_v|+|B_v|\le n_v.$$

$$d_u(u)+2d_u(v)\le 2n_u\text{ and }{d}_v(v)+2d_v(u)\le 2n_v.$$

$$2d_u(u)+2d_v(v)\le 3n_u-1\text{ and }2d_v(v)+2d_v(u)\le 3n_v-1.$$

$$2d(u)+2d(v)=(2d_u(u)+2d_v(u))+(2d_v(v)+2d_u(v))\le 3(n_u+n_v)-2=3n-2,$$

Full text

**Keywords: **Spanning Tree, Degree sequence, Arboreal.

††Partially supported by Conacyt, México.

1. Introduction

O. Ore [3] proved that if $G$ is a graph with $n$ vertices such that $d(u)+d(v)\geq n-1$ for each pair $u,v$ of non-adjacent vertices, then $G$ contains a hamiltonian path. This result has been generalized in many directions.

S. Win [5] showed that if $r\geq 2$ is an integer and $G$ is a connected graph with $n$ vertices such that $d(u_{1})+d(u_{2})+\cdots+d(u_{r})\geq n-1$ for each set of $r$ independent vertices of $G$, then $G$ has a spanning tree with maximum degree at most $r$.

Years later, H. Broersma and H. Tuinstra [2] showed that if $s\geq 2$ is an integer and $G$ is a connected graph with $n$ vertices such that $d(u)+d(v)\geq n-s+1$ for each pair $u,v$ of non-adjacent vertices, then $G$ contains a spanning tree with at most $s$ vertices with degree 1.

E. Rivera-Campo [4] gave a condition on the graph $G$ that bounds the degree of each vertex in a certain spanning tree $T$ of $G$ and the number of vertices of $T$ with degree 1.

Theorem 1**.**

Let $n$, $k$, and $d_{1},d_{2},\ldots,d_{n}$ be integers with $1\leq k\leq n-1$ and $2\leq d_{1}\leq d_{2}\leq\cdots\leq d_{n}\leq n-1$. If $G$ is a $k$-connected graph with vertex set $V(G)=\left\{w_{1},w_{2},\ldots,w_{n}\right\}$ such that $d(u)+d(v)\geq n-1-\sum_{j=1}^{n}{(d_{j}-2)}$ for each pair $u,v$ of non-adjacent vertices, then $G$ contains a spanning tree $T$ with at most $2+\sum_{j=1}^{n}{(d_{j}-2)}$ vertices with degree 1 and such that $d_{T}(w_{j})\leq d_{j}$ for $j=1,2,\ldots,n$.

Let $n$ be a positive integer. An arboreal sequence is a sequence of positive integers $d_{1},d_{2},\ldots,d_{n}$ such that $\sum_{j=1}^{n}{d_{j}}=2(n-1)$. It is well known that a sequence $\sigma=d_{1},d_{2},\ldots,d_{n}$ is arboreal if and only if there is a tree whose vertices have degrees $d_{1},d_{2},\ldots,d_{n}$.

Let $\sigma=d_{1},d_{2},\ldots,d_{n}$ be an arboreal sequence and $G$ be a labelled graph with vertex set $V(G)=\left\{w_{1},w_{2},\ldots,w_{n}\right\}$. A spanning tree $T$ of $G$ has degree sequence $\sigma$ if $d_{T}(w_{i})=d_{i}$ for $i=1,2,\ldots,n$. In this note we prove the following result:

Theorem 2**.**

Let $n\geq 4$ be an integer and $G$ be a labelled graph with vertex set $V(G)=\left\{w_{1},w_{2},\ldots,w_{n}\right\}$. If $d(u)+d(v)\geq\frac{3n-1}{2}$ for each pair $u,v$ of non-adjacent vertices, then $G$ contains a spanning tree $T$ with degree sequence $\sigma$ for each arboreal sequence $\sigma=d_{1},d_{2},\ldots,d_{n}$ with $1\leq d_{i}\leq 3$ for $i=1,2,\ldots,n$.

For each positive integer $k$ let $X_{k}=\{x_{1},x_{2},\ldots,x_{k}\}$, $Y_{k}=\{y_{1},y_{2},\ldots,y_{k}\}$ and $Z_{k}=\{z_{1},z_{2},\ldots,z_{2k+2}\}$ be pairwise disjoint sets of vertices and let $G_{k}$ be the complete graph with vertex set $X_{k}\cup Y_{k}\cup Z_{k}$ with all edges $x_{i}y_{i}$, $1\leq i,j\leq k$ removed. See Fig. 1 for the case $k=2$.

We claim that the graph $G_{k}$ contains no spanning tree $T$ such that $d_{T}(x_{i})=d_{T}(y_{i})=3$ for $i=1,2,\ldots,k$ and $d_{T}(z_{j})=1$ for $j=1,2,\ldots,2k+2$; for if $T$ is such a tree, then $T-Z_{k}$ would be a spanning tree of the subgraph $G\left[X_{k}\cup Y_{k}\right]$ of $G$, induced by the set $X_{k}\cup Y_{k}$, which is not possible since $G\left[X_{k}\cup Y_{k}\right]$ is not connected. On the other hand, if $u$ and $v$ are non-adjacent vertices of $G_{k}$, without loss of generality we may assume $u\in X_{k}$ and $v\in Y_{k}$. Therefore

[TABLE]

where $n=4k+2$ is the number of vertices of $G_{k}$. This shows that the degree-sum condition in Theorem 2 is tight.

Whenever possible we follow the notation of J. A. Bondy and U. S. R. Murty [1].

2. Proof of Theorem 2

Suppose the result is false. Then for certain integer $n\geq 4$ and certain arboreal sequence $\sigma=d_{1},d_{2},\ldots,d_{n}$ with $1\leq d_{i}\leq 3$ for $i=1,2,\ldots,n$ there exists a counterexample. That is a labelled graph $G$ with vertex set $V(G)=\left\{w_{1},w_{2},\ldots,w_{n}\right\}$ such that $G$ contains no spanning tree with degree sequence $\sigma$ while $d(u)+d(v)\geq\frac{3n-1}{2}$ for each pair $u,v$ of non-adjacent vertices of $G$. We choose $G$ with the maximum possible number of edges while remaining a counter example with $n$ vertices.

Since $\sigma$ is an arboreal sequence of order $n$, a counterexample cannot be a complete graph of order $n$. Let $u,v$ be non-adjacent vertices of $G$. By the choice of $G$, the graph $G+uv$ is not a counterexample and contains a spanning tree $T$ with degree sequence $\sigma$. Therefore $G$ contains a spanning forest $F=T-uv$ with exactly two components $T_{u}$ and $T_{v}$ with $u\in V(T_{u})$ and $v\in V(T_{v})$ such that $d_{F}(u)=d_{i}-1$, $d_{F}(v)=d_{j}-1$ and $d_{f}(w_{r})=d_{r}$ for each vertex $u\neq w_{r}\neq v$, where $i$ and $j$ are such that $u=w_{i}$ and $v=w_{j}$.

Orient the edges of $F$ in such a way that $T_{u}$ and $T_{v}$ become outdirected trees $\overrightarrow{T_{u}}$ and $\overrightarrow{T_{v}}$ with roots $u$ and $v$, respectively. For each vertex $u\neq w\neq v$ let $w^{-}$ denote the unique vertex of $G$ such that the edge $w^{-}w$ is oriented from $w^{-}$ to $w$ in $\overrightarrow{F}$. Let

[TABLE]

[TABLE]

Notice that $|A_{u}\cap B_{u}|=0$, for if $z^{-}\in A_{u}\cap B_{u}$, then $(F-z^{-}z)+\{uz,vz^{-}\}$ would be a spanning tree of $G$ with degree sequence $\sigma$, which is not possible (See Fig. 2). Analogously $|A_{v}\cap B_{v}|=0$ and therefore

[TABLE]

where $n_{u}$ and $n_{v}$ are the number of vertices of $T_{u}$ and $T_{v}$, respectively.

In an abuse of notation, for each vertex $x$ of $G$ we denote by $d_{u}(x)$ and $d_{v}(x)$ the number of vertices of $T_{u}$ and $T_{v}$, respectively, which are adjacent to $x$ in $G$. Clearly

[TABLE]

Also notice that

[TABLE]

since the out-degree of each vertex of $\overrightarrow{F}$ is at most 2. Then

[TABLE]

Therefore

[TABLE]

Since $u$ is not adjacent to $u$ and $v$ is not adjacent to $v$ in $G$, $d_{u}(u)\leq n_{u}-1$ and $d_{v}(v)\leq n_{v}-1$. Adding these to the previous inequalities we obtain

[TABLE]

These imply

[TABLE]

which is not possible since $d(u)+d(v)\geq\frac{3n-1}{2}.$ ∎

3. FINAL REMARKS

With the same approach, we can prove the following generalization of Theorem 2.

Theorem 3**.**

Let $r\geq 2$ be an integer and $G$ be a labelled graph with vertex set $V(G)=\left\{v_{1},v_{2},\ldots,v_{n}\right\}$ with $n\geq r+1$. If $d(u)+d(v)\geq\frac{(2r-3)n-(2r-5)}{r-1}$ for each pair $u$, $v$ of non-adjacent vertices, then $G$ has a spanning tree $T$ with degree sequence $\sigma$ for each arboreal sequence $\sigma=d_{1},d_{2},\ldots,d_{n}$ with $1\leq d_{i}\leq r$ for $i=1,2,\ldots,n$.

Let $r\geq 2$ be an integer. For each positive integer $k$ let $X_{k}=\left\{x_{1},x_{2},\ldots,x_{k}\right\},Y_{k}=\left\{y_{1},y_{2},\ldots,y_{k}\right\}$ and $Z_{k,r}=\left\{z_{1},z_{2},\ldots,z_{2k(r-2)+2}\right\}$ be pairwise disjoint vertex sets and let $G_{k,r}$ be the complete graph with vertex set $X_{k}\cup Y_{k}\cup Z_{k,r}$ with all edges $x_{i}y_{i},1\leq i,j\leq k$ removed. As for the graphs $G_{k}=G_{k,3}$ in the introduction, we claim that the graphs $G_{k,r}$ show that the degree-sum condition in Theorem 3 is also tight.