We show that all invertible n × n matrices over any finite field 𝔽_q can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or dividing a row by the generator of the multiplicative group of 𝔽_q. This even holds in the more general setting where the pairs of rows that can be added or subtracted are specified by an arbitrary transition tree that has to satisfy some mild constraints. Moreover, we can prescribe the first and the last matrix if n ≥ 3, or n = 2 and q > 2. In other words, the corresponding flip graph on all invertible n × n matrices over 𝔽_q is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs.

Abstract. In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. First, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage–Squire–West construction with a recent algorithm for generating elimination trees of chordal graphs. Second, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung–Hoang–Mütze–Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage–Squire–West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length 2(n+1) with exactly n+1 many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book The Art of Computer Programming, Knuth raised a stronger form of this conjecture (Problem 56 in section 7.2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has 2n+1 blocks of the same length, and each block is obtained by adding s=1 (modulo 2n+1) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to 2n+1. We also present an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time, using $\mathcal{O}(n)$ memory in total.

The well-known middle levels conjecture asserts that for every integer n ≥ 1, all binary strings of length 2(n + 1) with exactly n + 1 many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book ‘The Art of Computer Programming Vol. 4A’ Knuth raised a stronger form of this conjecture (Problem 56 in Section 7.2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has 2n + 1 blocks of the same length, and each block is obtained by adding s = 1 (modulo 2n + 1) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts s ≥ 1 that are coprime to 2n + 1. We also present an algorithm to compute this ordering, generating each new bitstring in (n) time, using (n) memory in total.