Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

TL;DR

This paper establishes improved bounds for the ultimate independence ratio of odd wheel graphs, advancing understanding of their independence properties in Cartesian graph powers.

Contribution

It provides the first tight upper bounds for odd wheels' independence ratios, including the case W_5, using combinatorial and computational methods.

Findings

Upper bound for odd wheels of length ≥7: less than 1/3.

Improved upper bound for W_5: 1019/3888.

Combines counting, recursive bounds, and computer calculations.

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.