Hamiltonian paths, unit-interval complexes, and 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 Pósa'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 (tight and weak) Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1)Every unit-interval strongly-connected d-dimensional simplicial complex is traceable.(This extends the well-known result “unit-interval connected graphs are traceable”.)(2)Every unit-interval 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 Gröbner 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.