Tighter Bounds on the Independence Number of the Birkhoff Graph

The Birkhoff graph 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 α ( B n ) of B n, namely, we show that α ( B n ) ≤ O ( n ! / 1 . 9 7 n ) improving on the previous known bound of α ( B n ) ≤ O ( n ! / 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 α ( B n ) by a factor of n / 2. This construction is based on a new proper coloring of B n, which also gives an upper bound on the chromatic number χ ( B n ) of B n. Via known connections, the upper bound on α ( B n ) implies alphabet size lower bounds for a family of maximally recoverable codes on grid-like topologies.