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.
Categories: math.CO