Sufficient conditions for a graph with minimum degree to have a component factor
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2025
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| _version_ | 1866909572200923136 |
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| author | Wu, Jie |
| author_facet | Wu, Jie |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for a graph with minimum degree to have a component factor Wu, Jie Combinatorics 05C50, 05C70, 90B99 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. |
| title | Sufficient conditions for a graph with minimum degree to have a component factor |
| topic | Combinatorics 05C50, 05C70, 90B99 |
| url | https://arxiv.org/abs/2504.06619 |