The Lower and Upper Bounds of Turán Number for Odd Wheels

Abstract

The Turán number for a graph H, denoted by $\text {ex}(n,H)$, is the maximum number of edges in any simple graph with n vertices which doesn’t contain H as a subgraph. In this paper we find the lower and upper bounds for $\text { ex}(n,W_{2t+1})$. We show that if $n\ge 4t$, then $\text { ex}(n,W_{2t+1})\ge \left\lfloor \lfloor \frac{2n+t}{4}\rfloor (n+\frac{t-1}{2}-\lfloor \frac{2n+t}{4}\rfloor )\right\rfloor +1.$ We also show that for sufficiently large n and $t\ge 5$, $\text { ex}(n,W_{2t+1})\le \frac{ n^2 }{4}+{t-1\over 2}n$. Moreover we find the exact value of the Turán number for $W_9$. That is, we show that for sufficiently large n, $\text { ex}(n,W_9)= \lfloor \frac{n^2}{4}\rfloor +\lceil \frac{3}{4}n\rceil +1$.