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Cliques and independent subgroups of the Birkhoff polytope graph

The Birkhoff polytope $Ω_n$ is the polytope of doubly stochastic matrices of order $n$. The Birkhoff polytope graph $G(Ω_n)$ is the skeleton of $Ω_n$; it is the Cayley graph whose vertex set consists of the elements of the symmetric group ${\rm Sym}(n)$ of degree $n$, where two permutations are adjacent if one equals the product of the other with a cycle. We study the combinatorial structure of this graph, focusing on its maximal and maximum cliques and on its independent subgroups (subgroups of ${\rm Sym}(n)$ whose elements are pairwise nonadjacent in the graph). We obtain maximal subgroups of $G(Ω_n)$ and establish both a lower bound and an upper bound for its clique number. Especially, we prove that if $K$ is a subset of ${\rm Sym}(n)$ consisting of 3-cycle permutations such that $δ_1^{-1}δ_2$ is a single cycle for all $δ_1,δ_2\in K$, then the maximum size of $K$ is $\lfloor (n-1)^2/4\rfloor$, which can be viewed as an Erdős-Ko-Rado-type theorem for ${\rm Sym}(n)$.