Taylor Ball | Searcharxiv

arXiv2020

Coding theory package for Macaulay2

In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We define an object {\it evaluation code}, a construction which allows to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families of linear codes such as Hamming codes, cyclic codes, Reed--Solomon codes, Reed--Muller codes, Cartesian codes, monomial--Cartesian codes, and toric codes. In addition, we define functions for the syndrome decoding algorithm and locally recoverable code construction, which are important tools in applications of linear codes. The package \textit{CodingTheory.m2} is available at \url{https://github.com/Macaulay2/Workshop-2020-Cleveland/tree/CodingTheory/CodingTheory}

cs.IT↗

arXiv2019

Independent set and matching permutations

Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $\pi$ of $\{1, \ldots, m\}$ is an {\em independent set permutation} of $G$ if $$ a_{\pi(1)}(G) \leq a_{\pi(2)}(G) \leq \cdots \leq a_{\pi(m)}(G) $$ where $a_k(G)$ is the number of independent sets of size $k$ in $G$. In 1987 Alavi, Malde, Schwenk and Erd\H{o}s proved that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $\alpha(G)=m$, i.e. with largest independent set having size $m$. They raised the question of determining, for each $m$, the smallest number $f(m)$ such that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $\alpha(G)=m$ and with at most $f(m)$ vertices, and they gave an upper bound on $f(m)$ of roughly $m^{2m}$. Here we settle the question, determining $f(m)=m^m$, and make progress on a related question, that of determining the smallest order such that every permutation of $\{1, \ldots, m\}$ is the {\em unique} independent set permutation of some graph of at most that order. More generally we consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on $\{1, \ldots, m\}$ can be realized by the independent set sequence of some graph with $\alpha(G)=m$ and with at most $m^{m+2}$ vertices. Alavi et al. also considered {\em matching permutations}, defined analogously to independent set permutations. They observed that not every permutation of $\{1,\ldots,m\}$ is a matching permutation of some graph with largest matching having size $m$, putting an upper bound of $2^{m-1}$ on the number of matching permutations of $\{1,\ldots,m\}$. Confirming their speculation that this upper bound is not tight, we improve it to $O(2^m/\sqrt{m})$.

math.CO↗

arXiv2015

On the cop number of generalized Petersen graphs

We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some families of generalized Petersen graphs. We also extend these ideas to prove that the cop number of any connected I-graph is at most 5.

math.CO↗