Mathew, R. and Newman, I. and Rabinovich, Y. and Rajendraprasad, D.
(2021)
Hamiltonian and pseudoHamiltonian cycles and fillings in simplicial complexes.
Journal of Combinatorial Theory. Series B, 150.
pp. 119143.
ISSN 00958956
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Abstract
We introduce and study a ddimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian ddimensional cycles in Knd (the complete simplicial dcomplex over a vertex set of size n). Hamiltonian dcycles are the simple dcycles of a complete rank, or, equivalently, of size 1+(n−1d). The discussion is restricted to the fields F2 and Q. For d=2, we characterize the n's for which Hamiltonian 2cycles exist. For d=3 it is shown that Hamiltonian 3cycles exist for infinitely many n's. In general, it is shown that there always exist simple dcycles of size (n−1d)−O(nd−3). All the above results are constructive. Our approach naturally extends to (and in fact, involves) dfillings, generalizing the notion of Tjoins in graphs. Given a (d−1)cycle Zd−1∈Knd, F is its dfilling if ∂F=Zd−1. We call a dfilling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size (n−1d). If a Hamiltonian dcycle Z over F2 contains a dsimplex σ, then Z∖σ is a Hamiltonian dfilling of ∂σ (a closely related fact is also true for cycles over Q). Thus, the two notions are closely related. Most of the above results about Hamiltonian dcycles hold for Hamiltonian dfillings as well. © 2021 Elsevier Inc.
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