Turán-type problems on $[a,b]$-factors of graphs, and beyond
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909403627651072 |
|---|---|
| author | Hao, Yifang Li, Shuchao |
| author_facet | Hao, Yifang Li, Shuchao |
| contents | Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the maximum size (resp. spectral radius) of an $n$-vertex $\mathcal{H}$-free graph. Denote by $\text{Ex}(n, \mathcal{H})$ the set of all $n$-vertex $\mathcal{H}$-free graphs with $\text{ex}(n, \mathcal{H})$ edges. Similarly, let $\mathrm{Ex}_{sp}(n,\mathcal{H})$ be the set of all $n$-vertex $\mathcal{H}$-free graphs with spectral radius $\text{ex}_{sp}(n, \mathcal{H})$. For positive integers $a, b$ with $a\leqslant b$, an $[a,b]$-factor of a graph $G$ is a spanning subgraph $F$ of $G$ such that $a\leqslant d_F(v)\leqslant b$ for all $v\in V(G)$, where $d_F(v)$ denotes the degree of the vertex $v$ in $F.$ Let $\mathcal{F}_{a,b}$ be the set of all the $[a,b]$-factors of an $n$-vertex complete graph $K_n$. In this paper, we determine the Turán number $\text{ex}(n,\mathcal{F}_{a,b})$ and the spectral Turán number $\text{ex}_{sp}(n,\mathcal{F}_{a,b}),$ respectively. Furthermore, the bipartite analogue of $\text{ex}(n,\mathcal{F}_{a,b})$ (resp. $\text{ex}_{sp}(n,\mathcal{F}_{a,b})$) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that $\mathrm{Ex}_{sp}(n,\mathcal{F}_{a,b})\subseteq \text{Ex}(n, \mathcal{F}_{a,b})$ holds for graphs and bipartite graphs. This partially answers an open problem proposed by Liu and Ning \cite{LN2023}. Our results may deduce a main result of Fan and Lin \cite{FL2022}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Turán-type problems on $[a,b]$-factors of graphs, and beyond Hao, Yifang Li, Shuchao Combinatorics 05C50 Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the maximum size (resp. spectral radius) of an $n$-vertex $\mathcal{H}$-free graph. Denote by $\text{Ex}(n, \mathcal{H})$ the set of all $n$-vertex $\mathcal{H}$-free graphs with $\text{ex}(n, \mathcal{H})$ edges. Similarly, let $\mathrm{Ex}_{sp}(n,\mathcal{H})$ be the set of all $n$-vertex $\mathcal{H}$-free graphs with spectral radius $\text{ex}_{sp}(n, \mathcal{H})$. For positive integers $a, b$ with $a\leqslant b$, an $[a,b]$-factor of a graph $G$ is a spanning subgraph $F$ of $G$ such that $a\leqslant d_F(v)\leqslant b$ for all $v\in V(G)$, where $d_F(v)$ denotes the degree of the vertex $v$ in $F.$ Let $\mathcal{F}_{a,b}$ be the set of all the $[a,b]$-factors of an $n$-vertex complete graph $K_n$. In this paper, we determine the Turán number $\text{ex}(n,\mathcal{F}_{a,b})$ and the spectral Turán number $\text{ex}_{sp}(n,\mathcal{F}_{a,b}),$ respectively. Furthermore, the bipartite analogue of $\text{ex}(n,\mathcal{F}_{a,b})$ (resp. $\text{ex}_{sp}(n,\mathcal{F}_{a,b})$) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that $\mathrm{Ex}_{sp}(n,\mathcal{F}_{a,b})\subseteq \text{Ex}(n, \mathcal{F}_{a,b})$ holds for graphs and bipartite graphs. This partially answers an open problem proposed by Liu and Ning \cite{LN2023}. Our results may deduce a main result of Fan and Lin \cite{FL2022}. |
| title | Turán-type problems on $[a,b]$-factors of graphs, and beyond |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2411.16143 |