The HypercubeGraph(n) command creates the hypercube graph of dimension n on $2^n$ vertices. The vertex labels are strings of binary vectors of length n, and two vertices are joined by an edge if and only if they differ in exactly one coordinate. Note, the hypercube graph for n=2 is a square and for n=3 it is a cube.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$H≔\mathrm{HypercubeGraph}\left(3\right)$

$H≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 12\; edge(s)}$

$\mathrm{Vertices}\left(H\right)$

$\left[''000''\,''001''\,''010''\,''011''\,''100''\,''101''\,''110''\,''111''\right]$

$\mathrm{Neighbors}\left(H\,''010''\right)$

$\left[''000''\,''011''\,''110''\right]$

$\mathrm{DrawGraph}\left(H\right)$

$\mathrm{IsHamiltonian}\left(H\,'C'\right)$

$\mathrm{true}$

$C$

$\left[''000''\,''100''\,''110''\,''010''\,''011''\,''111''\,''101''\,''001''\,''000''\right]$

$\mathrm{HighlightTrail}\left(H\,C\,\mathrm{red}\right)$

$\mathrm{DrawGraph}\left(H\right)$