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- W3122237265 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) The minors defining the determinantal facet ideal of any almost-closed complex form a lex-Groebner basis. (This revises a recent theorem by Ene et al., and extends a result by Herzog and others.) (4) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to any diagonal monomial order. In positive characteristic, they are even Frobenius split. (This provides the largest known class of determinantal facet ideals that are radical.)" @default.
- W3122237265 created "2021-02-01" @default.
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- W3122237265 date "2021-01-22" @default.
- W3122237265 modified "2023-09-23" @default.
- W3122237265 title "Hamiltonian paths, closed complexes, and determinantal facet ideals" @default.
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