Exact values of rainbow Turán numbers for fan graphs and even wheel graphs

An edge-colored graph is called rainbow if all its edges have distinct colors. For a fixed graph $H$, the rainbow Turán number $\text{exstar}(n,H)$ is the maximum number of edges in a properly edge-colored graph with $n$ vertices that does not contain a rainbow subgraph isomorphic to $H$. A $t$-fan $\text{Ft}$ ($t\ge 2$) is a graph formed by $t$ triangles sharing a common vertex. A wheel graph $\text{Wn}$ is constructed by connecting a new vertex $x$ to all vertices of a cycle $\text{Cn}$ with $n$ vertices. Keevash, Mubayi, Sudakov, Verstraëte~({\small Combin. Probab. Comput., 2007}) showed that $$\text{exstar}(n,F_t)\ge \text{floor}\frac{n^2}{4}+(t-1)\text{floor}\frac{n}{2}$$ by constructing extremal graphs for $n\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}4)$. In this paper, we propose such a method: for a graph $H$, the upper bound of $\text{exstar}(n,H)$ can be analyzed using the rainbow Turán number of $H^{\mathrm{\prime }}$, where $H^{\mathrm{\prime }}$ is obtained by deleting one vertex from $H$. By applying this method, we determine the exact values of the rainbow Turán numbers for two families of graphs when the number of vertices $n$ is sufficiently large. Specifically, we obtain: (1) when $n>230t$, $$\text{exstar}(n,\text{Ft})=\text{floor}\frac{n^2}{4}+(t-1)\text{floor}\frac{n}{2}-\text{Dn},$$ where $\text{Dn}=1$ if $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, and $\text{Dn}=0$ otherwise; (2) for graphs $H$ satisfying $W_{2t}\subset H\subset \text{Kts}$ ($t\le s$), when $n$ is sufficiently large, $$\text{exstar}(n,H)=\{\begin{array}{ll}\text{floor}\frac{n^2}{4}+\text{floor}\frac{(t-1)n}{2}-\text{Dn}& \text{if }t\text{ is odd},\\ \text{floor}\frac{n^2}{4}+\text{floor}\frac{(t-1)n}{2}& \text{if }t\text{ is even}\mathrm{.}\end{array}$$