Related papers: Graham's number stable digits: An exact solution

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition, and adjacent twin errors. The three codes he presented are length 3-digit…

Let $r\geq 3$ be an integer and $G$ be a graph. Let $\delta(G), \Delta(G)$, $\alpha(G)$ and $\mu(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper…

In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer…

It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially…

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of…

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result…

A pebbling move on a graph $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph $G$, denoted by $f(G)$, is the least integer $n$ such that, however $n$ pebbles are located…

For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks…

This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is…

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In…

Let $G=(V(G),E(G))$ be a simple graph. A set $D\subseteq V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V(G)\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…

In this study, we find all Pell and Pell-Lucas numbers which are sums of three base 10 repdigits. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport…

We consider a minimizing variant of the well-known \emph{No-Three-In-Line Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest number of points in an $n\times n$~grid such that every grid point lies on a common line…

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We…

A set of vertices of a graph $G$ such that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$. If additionally, the set of vertices induces a connected subgraph of $G$ then the…

We propose a new algorithm, call Sam to determinate the existence of the solutions for the equation $x^3+y^3+z^3=n$ for a fixed value $n > 0$ unknown.

The biclique partition number of a graph \(G\), denoted \( \operatorname{bp}(G)\), is the minimum number of biclique subgraphs that partition the edge set of \(G\). The Graham-Pollak theorem states that the complete graph on \( n \)…

An integer-valued sequence $\pi=(d_1, \ldots, d_n)$ is {\em graphic} if there is a simple graph $G$ with degree sequence of $\pi$. We say the $\pi$ has a realization $G$. Let $Z_3$ be a cyclic group of order three. A graph $G$ is {\em…

We generalize Lewis's result about a central series associated with the vanishing off subgroup. We write $V_{1}=V(G)$ for the vanishing off subgroup of $G$, and $V_{i}=[V_{i-1},G]$ for the terms in this central series. Lewis proved that…