Turán-Type Problems on [a, b]-Factors of Graphs, and Beyond

Abstract

Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is $\mathcal{H}$-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 Tur\'an number $\text{ex}(n,\mathcal{F}_{a,b})$ and the spectral Tur\'an 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 (arXiv:2307.14629, 2023). Our results may deduce a main result of Fan and Lin (arXiv:2211.09304v1, 2021).