Sufficient conditions for $k$-factors and spanning trees of graphs

TL;DR

This paper establishes new spectral and clique-based conditions that guarantee the existence of $k$-factors and spanning trees with specific properties in graphs, improving and extending previous results in graph theory.

Contribution

It introduces a clique-based sufficient condition for $k$-factors and a spectral condition for spanning $k$-trees in graphs, extending prior work and providing tight bounds.

Findings

A clique-based condition guarantees $k$-factors in graphs with minimum degree $ ext{δ}$.

A spectral condition ensures the existence of spanning $k$-trees in $m$-connected graphs.

A spectral criterion for spanning trees with leaf degree at most $k$ is established.

Abstract

For any integer $k\geq1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper we prove a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph with minimum degree at least $\delta$, which improves the sufficient condition of O \cite{O2021} based on the number of edges. For any integer $k\geq2,$ a spanning $k$-tree of a connected graph $G$ is a spanning tree in which every vertex has degree at most $k$. Motivated by the technique of Li and Ning \cite{Li2016}, we present a tight spectral condition for an $m$-connected graph to have a spanning $k$-tree, which extends the result of Fan, Goryainov, Huang and Lin \cite{Fan2021} from $m=1$ to general $m$. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of leaves adjacent to $v$ in $T$ for any $v\in V(T)$. We provide a tight spectral condition for the existence of a spanning tree with leaf degree at most $k$ in a connected graph with minimum degree $\delta$, where $k\geq1$ is an integer.