Extremal problems on shadows and hypercuts in simplicial complexes
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by
Nati Linial, Ilan Newman, Yuval Peled, Yuri Rabinovich
2014
Abstract
Let F be an nvertex 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 n2 edges, then at least
n^2/4 edges are in its shadow and this is tight.
Equivalently, the largest number of edges an nvertex cut can have is
n^2/4. These notions have natural analogs in higher
ddimensional 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
2dimensional " Qalmosthypertrees" (defined below) with an empty
shadow. We also show that the shadow of an " F_2almosthypertree"
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 ddimensional F
_2almosthypertree whose shadow has density o_n(1).
Finally, we mention several intriguing open questions.
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article
Stage
submitted
Date 20140804
Version
v1
Language
en
^{?}
1408.0602v1
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