Extremal problems on shadows and hypercuts in simplicial complexes release_6zvel2hfunamlnxrmrwukd6ihu

by Nati Linial, Ilan Newman, Yuval Peled, Yuri Rabinovich

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Let F be an n-vertex forest. We say that an edge e∉ F is in the shadow of F if F∪{e} contains a cycle. It is easy to see that if F is "almost a tree", that is, it has n-2 edges, then at least n^2/4 edges are in its shadow and this is tight. Equivalently, the largest number of edges an n-vertex cut can have is n^2/4. These notions have natural analogs in higher d-dimensional simplicial complexes, graphs being the case d=1. The results in dimension d>1 turn out to be remarkably different from the case in graphs. In particular the corresponding bounds depend on the underlying field of coefficients. We find the (tight) analogous theorems for d=2. We construct 2-dimensional " Q-almost-hypertrees" (defined below) with an empty shadow. We also show that the shadow of an " F_2-almost-hypertree" cannot be empty, and its least possible density is Θ(1/n). In addition we construct very large hyperforests with a shadow that is empty over every field. For d> 4 even, we construct d-dimensional F _2-almost-hypertree whose shadow has density o_n(1). Finally, we mention several intriguing open questions.
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Date   2014-08-04
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arXiv  1408.0602v1
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