Tur\'an numbers for non-bipartite graphs and applications to spectral extremal problems

Given a graph family $\mathcal{H}$ with $\min_{H\in \mathcal{H}}\chi(H)=r+1\geq 3$. Let ${\rm ex}(n,\mathcal{H})$ and ${\rm spex}(n,\mathcal{H})$ be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all $\mathcal{H}$-free graphs of order $n$, respectively. Denote by ${\rm EX}(n,\mathcal{H})$ (resp. ${\rm SPEX}(n,\mathcal{H})$) the set of extremal graphs with respect to ${\rm ex}(n,\mathcal{H})$ (resp. ${\rm spex}(n,\mathcal{H})$). In this paper, we use a decomposition family defined by Simonovits to give a characterization of which graph families $\mathcal{H}$ satisfy ${\rm ex}(n,\mathcal{H})<e(T_{n,r})+\lfloor \frac{n}{2r} \rfloor$. Furthermore, we completely determine ${\rm EX}\big(n,\mathbb{G}(F_1,\ldots,F_k)\big)$ for $n$ sufficiently large, where $\mathbb{G}(F_1,\ldots,F_k)$ denotes a finite graph family which consists of $k$ edge-disjoint $(r+1)$-chromatic color-critical graphs $F_1,\ldots,F_k$. This result strengthens a theorem of Gy\H{o}ri, who settled the case that $F_1=\cdots =F_k = K_{r+1}$. Wang, Kang and Xue %[J. Combin. Theory Ser. B 159 (2023) 20--41] proved that ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for $n$ sufficiently large and any graph $H$ with ${\rm ex}(n,H)=e(T_{n,r})+O(1)$. As an application of our first theorem, we show that ${\rm SPEX}(n,\mathcal{H})\subseteq {\rm EX}(n,\mathcal{H})$ for $n$ sufficiently large and any finite family $\mathcal{H}$ with ${\rm ex}(n,\mathcal{H})<e(T_{n,r})+\lfloor \frac{n}{2r}\rfloor$. As an application of our second theorem we completely determine ${\rm SPEX}\big(n,\mathbb{G}(F_1,\ldots,F_k)\big)$ for $n$ sufficiently large. Finally, related problems are proposed for further research.