Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

Abstract

The ultimate independence ratio of a graph $G$ is defined as $\mathscr{I}(G) = \lim_{k\rightarrow\infty } \frac{\alpha(G^{\Box k})}{|V(G)|^k},$ where $\alpha(G^{\Box k})$ is the independence number of the Cartesian product of $k$ copies of $G$. For all graphs $G$, Hahn, Hell, and Poljak (1995) proved that $\frac{1}{\chi(G)} \leq \mathscr{I}(G) \leq \frac{1}{\omega(G)}$ where $\chi(G)$ is the chromatic number, and $\omega(G)$ is the clique number of $G$. So all graphs $G$ with $\chi(G) = \omega(G)$ satisfy $\mathscr{I}(G) = \frac{1}{\chi(G)} = \frac{1}{\omega(G)}$. A construction of Zhu demonstrates that there exists a graph $G$ with $\frac{1}{\chi(G)} < \mathscr{I}(G) < \frac{1}{\omega(G)}$, so neither equality holds in general. In response, Hahn, Hell, and Poljak conjectured that all wheel graphs $W_n$ satisfy $\mathscr{I}(W_n) = \frac{1}{\chi(W_n)}$. For even wheels $W_{2t}$ this follows from the fact $\chi(W_{2t}) = \omega(W_{2t}) = 3$. Odd wheels of length at least $5$ present a more challenging case, since $\chi(W_{2t+1}) = 4$ and $\omega(W_{2t+1}) = 3$. First, we prove that odd wheels of length at least $7$ satisfy $\mathscr{I}(W_{2t+1})\leq \frac{4t^2+6t}{3(2t+2)^2}<\frac{1}{3}$, which provides the best upper bound for large odd wheels. Next, we prove that $\mathscr{I}(W_5) \leq \frac{1019}{3888}$, improving an upper bound of Hahn, Hell, and Poljak that $\mathscr{I}(W_5) \leq \frac{11}{41}$. Our proofs combine counting arguments, recursive bounds on $\alpha(W^{\Box k}_{2t+1})$, and computer-assisted calculation in the $W_5$ case.