Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

TL;DR

This work analyzes the ultimate independence ratio $\mathscr{I}(G)$, with a focus on odd wheels $W_{2t+1}$. It develops a general bound $\mathscr{I}(G) \le \frac{\alpha(G^{\Box \ell}\Box K_k)}{k|V(G)|^{\ell}}$ and uses it to derive new upper bounds for odd wheels, notably $\mathscr{I}(W_{2t+1}) \le \frac{4t^2+6t}{3(2t+2)^2}$ for $t\ge3$ and $\mathscr{I}(W_5) \le \frac{1019}{3888}$, refining prior results. The paper also presents a comprehensive computational framework based on ILP/LP formulations to bound independence numbers and compute fractional chromatic numbers, applying it to obtain exact or near-exact values such as $\alpha(W_5^{\Box 3} \Box K_3)=170$, $\alpha(W_5^{\Box 4})\le 343$, and $\alpha(W_5^{\Box 4} \Box K_3) \le 1019$, which feed the improved bounds. By combining analytic bounds with computer-assisted proofs, the authors make substantial progress toward the conjecture $\mathscr{I}(W_{2t+1})=1/\chi(W_{2t+1})$ for odd wheels and provide a reusable methodology for exploring ultimate independence ratios in other graph families.

Abstract

The ultimate independence ratio of a graph $G$ is defined as $\mathscr{I}(G) = \lim_{k\rightarrow\infty } \frac{α(G^{\Box k})}{|V(G)|^k},$ where $α(G^{\Box k})$ is the independence number of the Cartesian product of $k$ copies of $G$. For all graphs $G$, Hahn, Hell, and Poljak (1995) proved that $\frac{1}{χ(G)} \leq \mathscr{I}(G) \leq \frac{1}{ω(G)}$ where $χ(G)$ is the chromatic number, and $ω(G)$ is the clique number of $G$. So all graphs $G$ with $χ(G) = ω(G)$ satisfy $\mathscr{I}(G) = \frac{1}{χ(G)} = \frac{1}{ω(G)}$. A construction of Zhu demonstrates that there exists a graph $G$ with $\frac{1}{χ(G)} < \mathscr{I}(G) < \frac{1}{ω(G)}$, so neither equality holds in general. In response, Hahn, Hell, and Poljak conjectured that all wheel graphs $W_n$ satisfy $\mathscr{I}(W_n) = \frac{1}{χ(W_n)}$. For even wheels $W_{2t}$ this follows from the fact $χ(W_{2t}) = ω(W_{2t}) = 3$. Odd wheels of length at least $5$ present a more challenging case, since $χ(W_{2t+1}) = 4$ and $ω(W_{2t+1}) = 3$. 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 $α(W^{\Box k}_{2t+1})$, and computer-assisted calculation in the $W_5$ case.

Figures (8)

• Figure 1: The branch and bound tree for assertion $(v)$ in Lemma \ref{['lemma:third_power_center_independence']}.
• Figure 2: The possible cases when showing solutions $v_3,v_4$ solutions are infeasible in assertion $(v)$ of Lemma \ref{['lemma:third_power_center_independence']}.
• Figure 3: The branch and bound tree for independent sets of size $57$ in $W^{\Box 3}_5$ where $|I_*| = 9$, as mentioned in in Lemma \ref{['lemma:57 center 9']}.
• Figure 4: An independent $S$ set of size $170$ in $W^{\Box 3}_5 \Box K_3$. Each set $S_i$ is represented by a colour in $W^{\Box 3}_5$, while $W^{\Box 3}_5$ is displayed using $6$ copies $W^{\Box 2}_5$ via the partition $I_{*},I_0,\dots, I_4$.
• Figure 5: A maximum independent set, which has size $29$, in $W^{\Box 2}_{5}\Box K_3$

Theorems & Definitions (37)

• Theorem 1.1: hahn1995ultimate
• Conjecture 1.2: hahn1995ultimate
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof : Proof of Theorem \ref{['thm:upper_bound_triangle']}
• Proposition 4.1