Show newer

On the Connectivity of Friends-and-strangers Graphs arxiv.org/abs/2410.21334

On the Connectivity of Friends-and-strangers Graphs

Friends-and-strangers graphs, coined by Defant and Kravitz, are denoted by $\mathsf{FS}(X,Y)$ where $X$ and $Y$ are both graphs on $n$ vertices. The graph $X$ represents positions and edges mark adjacent positions while the graph $Y$ represents people and edges mark friendships. The vertex set of $\mathsf{FS}(X,Y)$ consists of all one-to-one placements of people on positions, and there is an edge between any two placements if it is possible to swap two people who are friends and on adjacent positions to get from one placement to the other. Previous papers have studied when $\mathsf{FS}(X,Y)$ is connected. In this paper, we consider when $\mathsf{FS}(X,Y)$ is $k$-connected where a graph is $k$-connected if it remains connected after removing any $k-1$ or less vertices. We first consider $\mathsf{FS}(X,Y)$ when $Y$ is a complete graph or star graph. We find tight bounds on their connectivity, proving their connectivity equals their minimum degree. We further consider the size of the connected components of $\mathsf{FS}(X,\mathsf{Star}_n)$ where $X$ is connected. We show that asymptotically similar conditions as the conditions mentioned by Bangachev are sufficient for $\mathsf{FS}(X,Y)$ to be $k$-connected. Finally, we consider when $X$ and $Y$ are independent Erdős--Rényi random graphs on $n$ vertices and edge probability $p_1$ and $p_2,$ respectively. We show that for $p_0 = n^{-1/2+o(1)},$ if $p_1p_2\geq p_0^2$ and $p_1,$ $p_2 \geq w(n) p_0$ where $w(n) \rightarrow 0$ as $n \rightarrow \infty,$ then $\mathsf{FS}(X,Y)$ is $k$-connected with high probability. This is asymptotically tight as we show that below an asymptotically similar threshold $p_0'=n^{-1/2+o(1)}$, the graph $\mathsf{FS}(X,Y)$ is disconnected with high probability if $p_1p_2 \leq (p_0')^2$.

arXiv.org

Cryptarithmically unique terms in integer sequences arxiv.org/abs/2410.21427

Cryptarithmically unique terms in integer sequences

A cryptarithm (or alphametic) is a mathematical puzzle in which numbers are represented with words in such a way that identical letters stand for equal digits and distinct letters for unequal digits. An alphametic puzzle is usually given in the form of an equation that needs to be solved, such as SEND + MORE = MONEY. Alternatively, here we will consider cryptarithms constrained not by an equation but by a particular subsequence of natural numbers, for example perfect squares or primes. Such a cryptarithm has a unique solution if there is exactly one term in the sequence that has the corresponding pattern of digits. We will call such terms cryptarithmically unique. Here we estimate the density of such terms in an arbitrary sequence for which the overall density of terms among integers is known. In particular, among all perfect squares below 10^12, slightly less than one half are cryptarithmically unique, their density increasing toward larger numbers. Cryptarithmically unique prime numbers, however, are initially very scarce. Combinatorial estimates suggest that their density should drop below 10^-300 for decimal lengths of approximately 1829 digits, but then it recovers and is asymptotic to unity for very large primes. Finally, we introduce and discuss primonumerophobic digit patterns that no prime number happens to have.

arXiv.org
Show older
Qoto Mastodon

QOTO: Question Others to Teach Ourselves
An inclusive, Academic Freedom, instance
All cultures welcome.
Hate speech and harassment strictly forbidden.