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

Abstract

An edge-colored graph is called rainbow if all its edges have distinct colors. For a fixed graph $H$, the rainbow Turán number $\exstar(n, H)$ 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 $\Ft$ ($t\ge 2$) is a graph formed by $t$ triangles sharing a common vertex. A wheel graph $\Wn$ is constructed by connecting a new vertex $x$ to all vertices of a cycle $\Cn$ with $n$ vertices. Keevash, Mubayi, Sudakov, Verstraëte~({\small Combin. Probab. Comput., 2007}) showed that \[ \exstar(n, F_t) \geq \floor{\frac{n^2}{4}} + (t-1)\floor{\frac{n}{2}} \] by constructing extremal graphs for $n\not\equiv 2(\mathrm{mod}4)$. In this paper, we propose such a method: for a graph $H$, the upper bound of $\exstar(n, H)$ can be analyzed using the rainbow Turán number of $H^{\prime }$, where $H^{\prime }$ 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 $n>230t$, \[ \exstar(n, \Ft) = \floor{\frac{n^2}{4}} + (t-1)\floor{\frac{n}{2}} - \Dn, \] where $\Dn = 1$ if $n\equiv 2(\mathrm{mod}4)$, and $\Dn = 0$ otherwise; (2) for graphs $H$ satisfying $W_{2t} \subset H \subset \Kts$ ($t\le s$), when $n$ is sufficiently large, \[ \exstar(n, H) = \begin{cases} \floor{\frac{n^2}{4}} + \floor{\frac{(t-1)n}{2}} - \Dn & \text{if } t \text{ is odd}, \\ \floor{\frac{n^2}{4}} + \floor{\frac{(t-1)n}{2}} & \text{if } t \text{ is even}. \end{cases} \]

22 pages, 2 figures