Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

TL;DR

This paper explores high-dimensional generalizations of graph theory concepts like Hamiltonian paths and interval graphs, connecting them to algebraic structures such as determinantal facet ideals, and extends several classical results to simplicial complexes.

Contribution

It introduces a hierarchy of combinatorial properties for simplicial complexes that generalize interval and co-comparability graphs, linking these to algebraic properties of determinantal facet ideals.

Findings

Almost-closed strongly-connected complexes are traceable.

Certain complexes remain Hamiltonian after vertex deletion.

Unit-interval complexes have Groebner bases for their determinantal facet ideals.

Abstract

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are: (1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result "unit-interval connected graphs are traceable".) (2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that "unit-interval 2-connected graphs are Hamiltonian".) (3) Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Groebner basis for a diagonal term order which is compatible with the traceability of the complex. (This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa-Vandebogert.) (4) Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution. (This extends the result by Kiani and Saeedi-Madani that "only the complete graph has a binomial edge ideal with linear resolution".) (5) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.