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hypercube
Weak saturation of the cube in the clique ★
Determine .
Keywords: bootstrap percolation; hypercube; Weak saturation
Turán Problem for $10$-Cycles in the Hypercube ★★
Author(s): Erdos
![$ C_{10} $](/files/tex/11903dbe89daae659b28a8bead9b43cae2995c3d.png)
Keywords: cycles; extremal combinatorics; hypercube
Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation
Saturation in the Hypercube ★★
Author(s): Morrison; Noel; Scott
![$ 2\ell $](/files/tex/e6160c4357fdf2ec5854a3cc78837f8a67caa5c5.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
Keywords: cycles; hypercube; minimum saturation; saturation
Coloring squares of hypercubes ★★
Author(s): Wan
If is a simple graph, we let
denote the simple graph with vertex set
and two vertices adjacent if they are distance
in
.
![$ \chi(Q_d^{(2)}) = 2^{ \lfloor \log_2 d \rfloor + 1} $](/files/tex/926a7d4eb7afdf8197ff1121f017d56c061947b8.png)
Matchings extend to Hamiltonian cycles in hypercubes ★★
Keywords: Hamiltonian cycle; hypercube; matching
The Crossing Number of the Hypercube ★★
The crossing number of
is the minimum number of crossings in all drawings of
in the plane.
The -dimensional (hyper)cube
is the graph whose vertices are all binary sequences of length
, and two of the sequences are adjacent in
if they differ in precisely one coordinate.
![$ \displaystyle \lim \frac{cr(Q_d)}{4^d} = \frac{5}{32} $](/files/tex/8a347f2f5bc9527051396b3f7efcdc793f5c87f0.png)
Keywords: crossing number; hypercube
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