Tighter bounds on the independence number of the Birkhoff graph

The Birkhoff graph B-n 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 alpha(Bn) of Bn, namely, we show that alpha(B-n) <= O(n!/1.97n) improving on the previous known bound of alpha(B-n) <= O(n!/root 2( n)) 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 alpha(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 X(Bn) of Bn. Via known connections, the upper bound on alpha(Bn) implies alphabet size lower bounds for a family of maximally recoverable codes on grid-like topologies. (c) 2022 Elsevier Ltd. All rights reserved.