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

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 $\rho(G)$ 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.