2025/07/18 by Brenner, Sofia, Cardinal, Jean, McConville, Thomas +2 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.14327
For an arrangement H of hyperplanes in ℝn through the origin, a region is a connected subset of ℝn\setminusH. The graph of regions G(H) has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from H. We aim to compute a Hamiltonian path or cycle in the graph G(H), i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if H is a supersolvable arrangement, then the graph of regions G(H) has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions P(H,R0), obtained by orienting the graph G(H) away from a particular base region R0. Our second main result is that if H is supersolvable and R0 is a canonical base region, then for any lattice congruence ≡ on P(H,R0)=:L, the cover graph of the quotient lattice L/≡ has a Hamiltonian path. [...]