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Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. Motivated by results of Eckhoff and of Montejano and Oliveros on "tolerant" versions of Helly's theorem, we define the t-tolerance complex of K, T t (K) , as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t, d) such that, for every d-collapsible complex K, the t-tolerance complex T t (K) is h(t, d)-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem. [ABSTRACT FROM AUTHOR]

The d-dimensional algebraic connectivity ad(G) of a graph G = (V,E), introduced by Jordán and Tanigawa, is a quantitative measure of the d-dimensional rigidity of G that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set V into ℝd. Here, we analyze the d-dimensional algebraic connectivity of complete graphs. In particular, we show that, for d ≥ 3, ad(Kd+1) = 1, and for n ≥ 2d, ⌈ n 2 d ⌉ − 2 d + 1 ≤ a d (K n) ≤ 2 n 3 (d − 1) + 1 3. [ABSTRACT FROM AUTHOR]

Let X be a simplicial complex on vertex set V. We say that X is d-representable if it is isomorphic to the nerve of a family of convex sets in R d . We define the d-boxicity of X as the minimal k such that X can be written as the intersection of kd-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of X is a set τ ⊂ V such that τ ∉ X but σ ∈ X for any σ ⊊ τ . We prove that the d-boxicity of a simplicial complex on n vertices without missing faces of dimension larger than d is at most ⌊ n d / (d + 1) ⌋ . The bound is sharp: the d-boxicity of a simplicial complex whose set of missing faces form a Steiner (d , d + 1 , n) -system is exactly n d / (d + 1) . One of the main ingredients in the proof is the following bound on the representability of a complex: let V 1 , ... , V k be subsets of V such that V i ∉ X for all 1 ≤ i ≤ k , and for any missing face τ of X there is some 1 ≤ i ≤ k satisfying | τ \ V i | ≤ 1 . Then, X can be written as an intersection X = ⋂ i = 1 k X i , where, for all 1 ≤ i ≤ k , X i is a (| V i | - 1) -representable complex. In particular, X is (∑ i = 1 k (| V i | - 1)) -representable. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a graph and n a positive integer. Let In(G) be the abstract simplicial complex whose simplices are the subsets of V that do not contain an independent set of size n in G. We study the collapsibility numbers of the complexes In(G) for various classes of graphs, focusing on the class of graphs with maximum degree bounded by Δ. As an application, we obtain the following result Let G be a claw-free graph with maximum degree at most Δ. Then, every collection of ⌊ (Δ 2 + 1) (n − 1) ⌋ + 1 independent sets in G has a rainbow independent set of size n. [ABSTRACT FROM AUTHOR]

This research note explores the current issue of using podcasting as a resource for tourism marketing. It investigates the websites of Convention and Visitors Bureaus (CVB) in US cities for the use of podcasting to promote tourism. The findings show that only five CVBs currently use the technology of podcasting and the application is varied in form, approach, and content. Many more travel and destination podcasts exist separate from CVB sponsorship. The conclusions suggest that podcasting will become an important marketing tool for tourist destinations and merits study by tourism researchers and practitioners. [ABSTRACT FROM AUTHOR]