===== k-regular graph =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $n\in\mathbb N, n\ge 1$ |

| @#FFBB00: definiendum | @#FFBB00: $ Q_n\equiv\langle V,E \rangle $ |

| @#55EE55: postulate   | @#55EE55: $ V=\{0,1\}^n $ |

| @#FFFDDD: for all     | @#FFFDDD: $ v,w\in V $ |
| @#DDDDDD: range       | @#DDDDDD: $ k\in\mathbb N, 1\le k\ne n $ |

| @#55EE55: postulate   | @#55EE55: $ \{v,w\}\in E \leftrightarrow \exists! k.\ \pi_k(v)\neq \pi_k(w) $ |

==== Discussion ====
The n-cube $Q_n$ is the graph with vertices being n-tuples which are connected exactly if they differ by one coordinate. 

=== Examples ===
$V(Q_2)=\{\langle 0,0\rangle,\langle 0,1\rangle,\langle 1,0\rangle,\langle 1,1\rangle\}$

$E(Q_2)=\{\{\langle 0,0\rangle,\langle 0,1\rangle\},\{\langle 0,0\rangle,\langle 1,0\rangle\},\{\langle 0,1\rangle,\langle 1,1\rangle\},\{\langle 1,0\rangle,\langle 1,1\rangle\}\}$

... that's a square.
==== Parents ====
=== Subset of ===
[[Regular graph]]
=== Context ===
[[Cartesian product]]