Related papers: Verifying the Smallest Interesting Colour Code wit…
In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the…
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…
An index coding problem is called unicast-uniprior when each receiver demands a unique subset of messages while knowing another unique subset of messages apriori as side-information. In this work, we give an algorithm to compute the minrank…
As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain…
We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier results. Among other things,…
A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if…
This paper explores the use of large language models (LLMs), here represented by GPT 3.5-Turbo to perform coding for a thematic analysis. Coding is highly labor intensive, making it infeasible for most researchers to conduct exhaustive…
Decoding algorithms are essential to fault-tolerant quantum-computing architectures. In this perspective we explore decoding algorithms for the surface code; a prototypical quantum low-density parity-check code that underlies many of the…
This short note proposes a symbolic approach for representing and reasoning about quantum circuits using complex, vector or matrix-valued Boolean expressions. A major benefit of this approach is that it allows us to directly borrow the…
We give an introduction to the theory of quantum error correction using stabilizer codes that is geared towards the working computer scientists and mathematicians with an interest in exploring this area. To this end, we begin with an…
A $q$-\emph{equitable coloring} of a graph $G$ is a proper $q$-coloring such that the sizes of any two color classes differ by at most one. In contrast with ordinary coloring, a graph may have an equitable $q$-coloring but has no equitable…
Quantitative research relies heavily on coding, and coding errors are relatively common even in published research. In this paper, we examine whether individuals are more or less likely to check their code depending on the results they…
We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the…
The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can…
Quantum error correction is rapidly seeing first experimental implementations, but there is a significant gap between asymptotically optimal error-correcting codes and codes that are experimentally feasible. Quantum LDPC codes range from…
As the first useful Quantum Computers will be quantum simulators, here the minimum number of qubits necessary for the solution of the Schroedinger equation in simple test problems is evaluated. From the present preliminary results it…
We address a well-known problem in combinatorics involving the identification of counterfeit coins with a systematic approach. The methodology can be applied to cases where the total number of coins is exceedingly large such that brute…
Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively…
A variational model for reconstruction of damaged color images is studied, in particular in the case where only finitely many colors are admissible for the reconstructed image. An existence result and regularity properties of minimizers are…
A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of…