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Register a new userLarge Simple d-Cycles in Simplicial Complexes
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We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$s density. This generalizes a well-known classical result of ErdH{o}s and Gallai cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.
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We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} choose d} - O(n^{d-3})$. All the above results are constructive. Our approach naturally extends to (and in fact, involves) $d$-fillings, generalizing the notion of $T$-joins in graphs. Given a $(d-1)$-cycle $Z^{d-1} in K_n^d$, ~$F$ is its $d$-filling if $partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $sigma$, then $Zsetminus sigma$ is a a Hamiltonian $d$-filling of $partial sigma$ (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 $d$-cycles hold for Hamiltonian $d$-fillings as well.
A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined by Chen, Yau and Yeh (Discrete Math., 241(2001), 153-170).
Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $mu_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $mu_{k}(X)$ for $kgeq d$ and $mu_{d-1}(X)$. In particular, we establish the following vanishing result: If $mu_{d-1}(X)>(1-binom{k+1}{d}^{-1})n$, then $tilde{H}^{j}(X;mathbb{R})=0$ for all $d-1leq j leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martinez-Sandoval and Montejano for general position sets in matroids.
We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexes the links of simplexes and their intersections are also multi-parameter random simplicial complexes. Thirdly, we find conditions under which a multi-parameter random simplicial complex is connected and simply connected.
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