Interchange graphs of (0,1)-matrices are maximally Hamiltonian

Abstract

For integer vectors R,S let A(R,S) denote the class of (0,1)-matrices with row sum vector R and column sum vector S. Its interchange graph G(R,S) has A(R,S) as its vertex set, two matrices being adjacent when they differ by a single 2 x 2 interchange. Brualdi asked whether G(R,S) is Hamiltonian for every R,S. We prove the stronger statement that G(R,S) is maximally Hamiltonian: Hamilton-laceable when bipartite, and Hamilton-connected when not. The proof is a structural induction on the number of matrices in the class, organized by the structure theory of interchange graphs. Deleting inactive lines and splitting invariant positions expresses any class as a Cartesian product, reducing the argument to the prime factors. The bipartite classes are products of complete transposition graphs; we settle them together, without induction, by proving they are paired 2-disjoint-path-coverable and hence Hamilton-laceable, using a recent theorem of Coleman, Fischberg, Gong, Harrington and Wong on paired disjoint path covers. The non-bipartite classes divide into three cases: products assembled from smaller factors, a base of Johnson graphs and small classes, and the large prime classes, treated by a pivot-and-fiber construction whose line quotients are matroid base-exchange graphs. The complete argument has been machine-checked in the Lean 4 proof assistant from first principles together with seven cited results of the literature; the disjoint-path-cover results it imports are themselves proved within the formalization.