Tighter bounds on the independence number of the Birkhoff graph

Abstract

The Birkhoff graph Bn is the Cayley graph of the symmetric group Sn, where two permutations are adjacent if they differ by a single cycle. Our main result is a tighter upper bound on the independence number α(Bn) of Bn, namely, we show that α(Bn)≤O(n!/1.97n) improving on the previous known bound of α(Bn)≤O(n!/2n) by Kane et al. (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 α(Bn) by a factor of n/2. This construction is based on a new proper coloring of Bn, which also gives an upper bound on the chromatic number χ(Bn) of Bn. Via known connections, the upper bound on α(Bn) implies alphabet size lower bounds for a family of maximally recoverable codes on grid-like topologies.