arXiv subjects

Josh Hiller

Publications and source records attributed to Josh Hiller.

5 recordsLinked to original sources

arXiv2020

Algebraic Properties of a Hypergraph Lifting Map

Recent work in hypergraph Ramsey theory has involved the introduction of a "lifting map" that associates a certain $3$-uniform hypergraph to a given graph, bounding cliques in a predictable way. In this paper, we interpret the lifting map as a linear transformation. This interpretation allows us to use algebraic techniques to prove several structural properties of the lifting map, culminating in new lower bounds for certain $3$-uniform hypergraph Ramsey numbers.

math.CO↗

arXiv2020

Groups that have a Partition by Commuting Subsets

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions -- with exact calculations in some cases. Finally, we examine how the size of partitions with the minimum number of parts behaves with respect to the direct product.

math.GR↗

arXiv2020

Avoiding monochromatic sub-paths in uniform hypergraph paths and cycles

We present a recursive formula for the number of ways to color $j$ vertices blue in an r-uniform hyperpath of size $n$ while avoiding a blue monochromatic sub-hyperpath of length k. We use this result to solve the corresponding problem for $(r-1)$-tight r-uniform paths and loose r-uniform cycles. This generalizes some well known results from reliability engineering and analysis.

math.CO↗

arXiv2019

Minimally Connected Hypergraphs

Graphs and hypergraphs are foundational structures in discrete mathematics. They have many practical applications, including the rapidly developing field of bioinformatics, and more generally, biomathematics. They are also a source of interesting algorithmic problems. In this paper, we define a \textit{construction process} for minimally connected $r$-uniform hypergraphs, which captures the intuitive notion of building a hypergraph piece-by-piece, and a numerical invariant called the \textit{tightness}, which is independent of the construction process used. Using these tools, we prove some fundamental properties of minimally connected hypergraphs. We also give bounds on their chromatic numbers and provide some results involving edge colorings. We show that every connected $r$-uniform hypergraph contains a minimally connected spanning subhypergraph and provide a polynomial-time algorithm for identifying such a subhypergraph.

math.CO↗

arXiv2017

A short note on the order of the Zhang-Liu matrices over arbitrary fields

We give necessary and sufficient conditions for the Zhang-Liu matrices to be diagonalizable over arbitrary fields and provide the eigen-decomposition when it is possible. We use this result to calculate the order of these matrices over any arbitrary field. This generalizes a result of the second author.

math.CO↗