Papers › Tighter Bounds on the Independence Number of the Birkhoff Graph
Tighter Bounds on the Independence Number of the Birkhoff Graph
Leonardo Nagami Coregliano, Fernando Granha Jeronimo
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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!/√(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 α(ℬₙ) by a factor of n/2. This construction is based on a 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.
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