Extending Edge-Colorings of Complete Uniform Hypergraphs

Abstract

A hypergraph $V(\mathcal{G})$ is an ordered pair$\mathcal{G}=(V(\mathcal{G}),E(\mathcal{G}))$ where $V(\mathcal{G})$ is a set of vertices of $\mathcal{G}$ and $E(\mathcal{G})$ is a collection of edge multisets of $\mathcal{G}$. If the size of every edge in the hypergraph is equal, then we call it a uniform hypergraph. A \textit{complete $h$-uniform hypergraph}, written $K^h_n$, is a uniform hypergraph with edge sizes equal to $h$ and has $n$ vertices where the edges set is the collection of all $h$-elements subset of its vertex set (so the total number of the edges is $\binom{n}{h}$). A hypergraph is called \textit{regular} if the degree of all vertices is the same. An $r$-factorization of a hypergraph is a coloring of the edges of a hypergraph such that the number of times each element appears in each color class is exactly $r$. A partial $r$-factorization is a coloring in which the degree of each vertex in each color class is at most $r$. The main problem under consideration in this thesis is motivated by Baranyai's famous theorem and Cameron's question from 1976. Given a partial $r$-factorization of $K^h_m$, we are interested in finding the necessary and sufficient conditions under which we can extend this partial $r$-factorization to an $r$-factorization of $K^h_n$. The case $h=3$ of this problem was partially solved by Bahmanian and Rodger in 2012, and the cases $h=4,5$ were partially solved by Bahmanian in 2018. Recently, Bahmanian and Johnsen showed that as long as $n\geq (h-1)(2m-1)$, the obvious necessary conditions are also sufficient. In this thesis, we improve this bound for all $h\in \{6,7,\dots,89\}$. Our proof is computer-assisted.