#### Year of Publication

2019

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Richard Ehrenborg

#### Abstract

A tournament graph *G* is a vertex set *V* of size *n*, together with a directed edge set *E* ⊂ *V* × *V* such that (*i*, *j*) ∈ *E* if and only if (*j*, *i*) ∉ *E* for all distinct *i*, *j* ∈ *V* and (*i*, *i*) ∉ *E* for all *i* ∈ *V*. We explore the following generalization: For a fixed *k* we orient every *k*-subset of *V* by assigning it an orientation. That is, every facet of the (*k* − 1)-skeleton of the (*n* − 1)-dimensional simplex on *V* is given an orientation. In this dissertation we bound the number of compatible *k*-simplices, that is we bound the number of *k*-simplices such that its (*k* − 1)-faces with the already-specified orientation form an oriented boundary. We prove lower and upper bounds for all *k* ≥ 3. For *k* = 3 these bounds agree when the number of vertices *n* is *q* or *q* + 1 where *q* is a prime power congruent to 3 modulo 4. We also prove some lower bounds for values *k* > 3 and analyze the asymptotic behavior.

#### Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2019.203

#### Recommended Citation

Chandrasekhar, Karthik, "BOUNDING THE NUMBER OF COMPATIBLE SIMPLICES IN HIGHER DIMENSIONAL TOURNAMENTS" (2019). *Theses and Dissertations--Mathematics*. 63.

https://uknowledge.uky.edu/math_etds/63