Sufficient conditions for a graph with minimum degree to have a component factor

TL;DR

This work addresses the existence of a specific component factor $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$ in connected graphs with minimum degree $\delta$ by establishing tight spectral conditions. It introduces the factor family and proves two main results: a sharp adjacency spectral radius condition and a tight signless Laplacian spectral radius condition that ensure the factor. The proofs use a spectral-extremal approach, comparing the given graph to an explicit extremal join graph $K_{\delta}\vee(K_{n-\lfloor k\delta/r\rfloor-\delta-1}\cup(\lfloor k\delta/r\rfloor+1)K_1)$ and applying subgraph spectral monotonicity and the isolated-vertices criterion. The results identify the extremal graph as the only exception and contribute to the broader understanding of how spectral properties constrain component factors in graphs.

Abstract

Let $\mathcal{T}_{\frac{k}{r}}$ denote the set of trees $T$ such that $i(T-S)\leq\frac{k}{r}|S|$ for any $S\subset V(T)$ and for any $e\in E(T)$ there exists a set $S^{*}\subset V(T)$ with $i((T-e)-S^{*})>\frac{k}{r}|S^{*}|$, where $r<k$ are two positive integers. A $\{C_{2i+1},T:1\leq i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$-factor of a graph $G$ is a spanning subgraph of $G$, in which every component is isomorphic to an element in $\{C_{2i+1},T:1\leq i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$. Let $A(G)$ and $Q(G)$ denote the adjacency matrix and the signless Laplacian matrix of $G$, respectively. The adjacency spectral radius and the signless Laplacian spectral radius of $G$, denoted by $ρ(G)$ and $q(G)$, are the largest eigenvalues of $A(G)$ and $Q(G)$, respectively. In this paper, we study the connections between the spectral radius and the existence of a $\{C_{2i+1},T:1\leq i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$-factor in a graph. We first establish a tight sufficient condition involving the adjacency spectral radius to guarantee the existence of a $\{C_{2i+1},T:1\leq i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$-factor in a graph. Then we propose a tight signless Laplacian spectral radius condition for the existence of a $\{C_{2i+1},T:1\leq i<\frac{r}{k-r},T\in\mathcal{T}_{\frac{k}{r}}\}$-factor in a graph.