Tighter Bounds on the Independence Number of the Birkhoff Graph

TL;DR

This paper establishes a tighter upper bound on the independence number of the Birkhoff graph using advanced representation theory and linear programming, and improves the lower bound through explicit coloring constructions, impacting coding theory.

Contribution

It introduces a novel combination of higher-order representation techniques and linear programming to tighten bounds on the independence number of the Birkhoff graph, and provides improved bounds on its chromatic number.

Findings

Upper bound on $\alpha( ext{Birkhoff}_n)$ improved to $O(n!/1.97^n)$

Lower bound on $\alpha( ext{Birkhoff}_n)$ increased by a factor of $n/2$

New coloring-based construction bounds the chromatic number $\chi( ext{Birkhoff}_n)$

Abstract

The Birkhoff graph $\mathcal{B}_n$ is the Cayley graph of the symmetric group $S_n$, where two permutations are adjacent if they differ by a single cycle. Our main result is a tighter upper bound on the independence number $\alpha(\mathcal{B}_n)$ of $\mathcal{B}_n$, namely, we show that $\alpha(\mathcal{B}_n) \le O(n!/1.97^n)$ improving on the previous known bound of $\alpha(\mathcal{B}_n) \le O(n!/\sqrt{2}^{n})$ by [Kane-Lovett-Rao, FOCS 2017]. Our approach combines a higher-order version of their representation theoretic techniques with linear programming. With an explicit construction, we also improve their lower bound on $\alpha(\mathcal{B}_n)$ by a factor of $n/2$. This construction is based on a proper coloring of $\mathcal{B}_n$, which also gives an upper bound on the chromatic number $\chi(\mathcal{B}_n)$ of $\mathcal{B}_n$. Via known connections, the upper bound on $\alpha(\mathcal{B}_n)$ implies alphabet size lower bounds for a family of maximally recoverable codes on grid-like topologies.