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If $n>0$ is an integer, let $[n]=\{1,\ldots,n\}$. Let $S_n$ denote the set of all permutations (bijections) $\pi:[n]\to[n]$. For which positive integers $n$ is there a bijection $\Phi:[n!]\to S_n$ su …

A simple, undirected graph $G=(V,E)$ is said to be vertex-transitive if for all $a,b\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(a) = b$. If $G = (\omega, E)$ is vertex-tran …

We consider any non-negative integer as an ordinal, that is $0=\emptyset$ and $n=\{0,\ldots,n-1\}$ for every positive integer. Let $\{0,1\}^n$ denote the set of $\{0,1\}$-vectors of length $n$. Define …

For any set $X$, let $[X]^2=\{\{a,b\}:a\neq b \in X\}$. If $n\in\mathbb{N}$ is a positive integer, let $S_n$ denote the collection of bijections $\varphi:\{0,\ldots,n-1\}\to\{0,\ldots,n-1\}$. Let $E_n …

What is an example of a simple graph $G = (\{1,\ldots,n\}, E)$, where $n\in\mathbb{N}$ is a positive integer, with the following properties? There is a path in $G$ of length $n$, every vertex has at …

Suppose we have two finite disjoint sets $A, B \neq \emptyset$ such that $|A|$ and $|B|$ differ by at most $1$, and let $\Gamma = (A\cup B, E)$ where $E\subseteq \big\{\{a,b\}: a\in A, b\in B\big\}$ b …

Given an integer $k>1$, is there a connected bipartite graph $\Gamma = (A, B, E)$ where $A\cap B = \emptyset$ and $E \subseteq \big\{\{a, b\}:a\in A, b\in B\big\}$ such that $|A| = |B|$, $\text{de …

Motivation. My youngest son has a bike lock with dials, and he forgot the unlocking combination completely, except that he remembered that digit $0$ appeared somewhere in the combination. So it was my …

Motivation. In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the su …

Let $\mathbb{N}$ denote the set of positive integers, and consider the graph $(\mathbb{N}, E)$ where a set $\{a,b\}$ of two distinct positive integers belongs to $E$ if there is an integer $k>1$ such …

Informally asking, can we step through all permutations of the set $\{1,\ldots,n\}$ by just using transpositions? More formally: For any $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$ and let $S_n$ be t …

Motivation. Today my sons played a card game, in which a fixed number $n$ of cards was lying on the table. A move consists of adding an unused card to the cards on the table, and removing a card from …

Let $G = (V,E)$ be a simple, undirected graph. We say that $v\neq w\in V$ lie on a common cycle if there is an integer $n\geq 3$ and an injective graph homomorphism $f: C_n\to V$ such that $v,w\in \te …