A closure result on spanning $k$-trees of graphs with given minimum degree

TL;DR

This paper establishes a closure condition for the existence of spanning k-trees in graphs with specified minimum degree, linking the property to the addition of an edge between certain nonadjacent vertices.

Contribution

It provides a new closure result characterizing when a connected graph contains a spanning k-tree based on minimum degree and vertex pair conditions.

Findings

A spanning k-tree exists if and only if adding a specific edge preserves the spanning k-tree property.

The result applies to graphs with minimum degree at least 1 and involves a condition on the degrees of nonadjacent vertices.

The closure condition relates the sum of degrees of nonadjacent vertices to the existence of spanning k-trees.

Abstract

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected graph of order $n$ with minimum degree $\delta$. Let $u$ and $v$ be two nonadjacent vertices of $G$ satisfying $d_{G}(u)+d_{G}(v)\geq n-1-(k-2)\delta$. Then $G$ has a spanning $k$-tree if and only if $G+uv$ has a spanning $k$-tree.