  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.
