The exact Turán number of disjoint graphs– A generalization of Simonovits’ theorem, and beyond

For a given graph $H$, we say that a graph $G$ is $H$-free if it does not contain $H$ as a subgraph. Let $\text{ex}(n,H)$ (resp. ${\text{ex}}_{sp}(n,H)$) denote the maximum size (resp. spectral radius) of an $n$-vertex $H$-free graph, and $\text{Ex}(n,H)$ (resp. ${\text{Ex}}_{sp}(n,H)$) denote the set of all $n$-vertex $H$-free graphs with $\text{ex}(n,H)$ edges (resp. spectral radius ${\text{ex}}_{sp}(n,H)$). We call $\text{ex}(n,H)$ (resp. ${\text{ex}}_{sp}(n,H)$) the Turán number (resp. spectral Turán number) of $H$. Suppose that we know the exact values of Turán numbers of $G_1,\dots ,G_k$, respectively. Can we get the exact value of the Turán number of the disjoint union of $G_1\cup \cdots \cup G_k$? Moon considered the disjoint union of complete graphs. A graph $G$ is color-critical if there exists an edge $e$ such that $\chi (G-e)<\chi \left(G\right)$. Simonovits extended Moon’s result to the disjoint union of color-critical graphs for sufficiently large $n$. Erdős et al. determined the Turán number of triangles sharing exactly one vertex. Chen et al. extended the result to complete graphs sharing exactly one vertex. Let $F$ be a disjoint union of $F_i$, where $F_i$ is a graph obtained by joining a vertex to all vertices of ${\bigcup }_j{F}_{i_j}$ and each $F_{i_j}$ is a color-critical graph. For a large $n$, we determined $\text{ex}(n,F)$ and ${\text{ex}}_{sp}(n,F)$. Moreover, we show that for each $F$ of these graphs, ${\text{Ex}}_{sp}(n,F)\subseteq \text{Ex}(n,F)$ provided $n$ is sufficiently large, which provides a large class of graphs that gives a positive answer to an open problem proposed recently by Liu and Ning: Characterize graphs $F$ satisfying ${\mathrm{Ex}}_{sp}(n,F)\subseteq \mathrm{Ex}(n,F)$.