Bryce Frederickson
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View article: Improved Decomposition Bounds for Partition Polytopes and Odd-Covers
Improved Decomposition Bounds for Partition Polytopes and Odd-Covers Open
The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $k_1,\dots,k_n$ can be encoded as the vertices of the partition polytope $\mathrm{PP}(k_1,\dots,k_n)$. We prove that, if $K = \max\{k_1,\dots,k_n\}$, then the comb…
View article: Towards Graham's rearrangement conjecture via rainbow paths
Towards Graham's rearrangement conjecture via rainbow paths Open
We study an old question in combinatorial group theory which can be traced back to a conjecture of Graham from 1971. Given a group $Γ$, and some subset $S\subseteq Γ$, is it possible to permute $S$ as $s_1, s_2, \ldots, s_d$ so that the pa…
View article: Treewidth, Hadwiger Number, and Induced Minors
Treewidth, Hadwiger Number, and Induced Minors Open
Treewidth and Hadwiger number are two of the most important parameters in structural graph theory. This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor. To formalise this, we say t…
View article: Regular bipartite decompositions of pseudorandom graphs
Regular bipartite decompositions of pseudorandom graphs Open
In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite dec…
View article: A note on the induced Ramsey theorem for spaces
A note on the induced Ramsey theorem for spaces Open
The aim of this note is to give a simplified proof of the induced version of the Ramsey theorem for vector spaces first proved by H. J. Prömel in 1986.Mathematics Subject Classifications: 05D10, 15A03Keywords: Ramsey theory, vector spaces
View article: Circuit decompositions of binary matroids
Circuit decompositions of binary matroids Open
Given a simple Eulerian binary matroid $M$, what is the minimum number of disjoint circuits necessary to decompose $M$? We prove that $|M| / (\operatorname{rank}(M) + 1)$ many circuits suffice if $M = \mathbb F_2^n \setminus \{0\}$ is the …
View article: Path Odd-Covers of Graphs
Path Odd-Covers of Graphs Open
We introduce and study "path odd-covers", a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The "path odd-cover number" $p_2(G)$ of a graph $G$ is the minimum cardinality of a collecti…
View article: Triangle Percolation on the Grid
Triangle Percolation on the Grid Open
We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subsete…
View article: Upper bounds on Ramsey numbers for vector spaces over finite fields
Upper bounds on Ramsey numbers for vector spaces over finite fields Open
For $B \subseteq \mathbb F_q^m$, let $\exaff(n,B)$ denote the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb …
View article: A note on the induced Ramsey theorem for spaces
A note on the induced Ramsey theorem for spaces Open
The aim of this note is to give a simplified proof of the induced version of the Ramsey theorem for vector spaces first proved by H. J. Prömel.
View article: Demystification of Graph and Information Entropy
Demystification of Graph and Information Entropy Open
Shannon entropy is an information-theoretic measure of unpredictability in probabilistic models. Recently, it has been used to form a tool, called the von Neumann entropy, to study quantum mechanics and network flows by appealing to algebr…
View article: On splitting and splittable families
On splitting and splittable families Open
A set $A$ is said to split a finite set $B$ if exactly half the elements of $B$ (up to rounding) are contained in $A$. We study the dual notions: (1) splitting family, which is a collection of sets such that any subset of $\{1,\ldots,k\}$ …