Cliques and independent subgroups of the Birkhoff polytope graph
Zejun Huang, Chi-Kwong Li, Eric Swartz, Nung-Sing Sze
TL;DR
This paper studies the cliques and independent subgroups of the Birkhoff polytope graph $G(\Omega_n)$, the Cayley graph of $\mathrm{Sym}(n)$ with generator set of cycles. It establishes both lower and upper bounds on the clique number $\omega(n)$ and fully characterizes maximum 2-cycle and 3-cycle cliques in many regimes, including an Erdős–Ko–Rado-type bound for 3-cycles: if $K \subseteq \mathrm{Sym}(n)$ has the property that $\delta_1^{-1}\delta_2$ is a single cycle for all $\delta_1,\delta_2 \in K$, then $|K| \le \lfloor (n-1)^2/4\rfloor$. It also constructs maximal independent subgroups of $\mathrm{Sym}(n)$, notably for even $n=2k$ the independent set $G_n$ with $|G_n|=k!2^{k-1}$, and proves maximality (every nontrivial coset contains a cycle). The results connect to extremal set theory and point to extensions toward transportation polytopes and higher-order doubly stochastic tensors as future directions.
Abstract
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 ${\rm Sym}(n)$.