A note on the Turán number of disjoint union of wheels

Abstract

The Turán number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C_{n-1}$. Let $m{W}_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Turán number and all extremal graphs for $m{W}_{2k+1}$. We also provide the Turán number and all extremal graphs for $W^h:=\bigcup _{i=1}^m{W}_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ and $h>0$.