Discoveries / combinatorics

Brualdi's Question on Hamiltonicity of Interchange Graphs

The interchange graph $G(R,S)$ has the $(0,1)$-matrices with row sums $R$ and column sums $S$ as vertices, adjacent when they differ by a single $2\times 2$ interchange. Brualdi asked whether $G(R,S)$ is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.