Extending Edge-Colorings of Complete Uniform Hypergraphs

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\u27s famous theorem and Cameron\u27s 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

Ryser Type Conditions for Extending Colorings of Triples

In 1951, Ryser showed that an $n\times n$ array $L$ whose top left $r\times s$ subarray is filled with $n$ different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order $n$ if and only if the number of occurrences of each symbol in $L$ is at least $r+s-n$. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let $X,Y$ be finite sets with $X\subsetneq Y$ and $|Y|\equiv 0 \pmod 3$. When can we extend a (proper) coloring of $\lambda \binom{X}{3}$ (all triples on a ground set $X$, each one being repeated $\lambda$ times) to a coloring of $\lambda \binom{Y}{3}$ using the fewest possible number of colors? It is necessary that the number of triples of each color in $\binom{X}{3}$ is at least $|X|-2|Y|/3$. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483--495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using H\"aggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295--313), we show that the number of triples of each color being at least $|X|/2-|Y|/6$ is sufficient. Finally we prove an Evans type result by showing that if $|Y|\geq 3|X|$, then any $q$-coloring of any subset of $\lambda \binom{X}{3}$ can be embedded into a $\lambda\binom{|Y|-1}{2}$-coloring of $\lambda \binom{Y}{3}$ as long as $q\leq \lambda \binom{|Y|-1}{2}-\lambda \binom{|X|}{3}/\lfloor{|X|/3}\rfloor$.Comment: 10 page

Bus interconnection networks

AbstractIn bus interconnection networks every bus provides a communication medium between a set of processors. These networks are modeled by hypergraphs where vertices represent the processors and edges represent the buses. We survey the results obtained on the construction methods that connect a large number of processors in a bus network with given maximum processor degree Δ, maximum bus size r, and network diameter D. (In hypergraph terminology this problem is known as the (Δ,D, r)-hypergraph problem.)The problem for point-to-point networks (the case r = 2) has been extensively studied in the literature. As a result, several families of networks have been proposed. Some of these point-to-point networks can be used in the construction of bus networks. One approach is to consider the dual of the network. We survey some families of bus networks obtained in this manner. Another approach is to view the point-to-point networks as a special case of the bus networks and to generalize the known constructions to bus networks. We provide a summary of the tools developed in the theory of hypergraphs and directed hypergraphs to handle this approach