# hypercube

## Weak saturation of the cube in the clique ★

Author(s): Morrison; Noel

Problem

Determine .

## Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

Problem   Bound the extremal number of in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

## Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

Author(s): Morrison; Noel

Problem   Determine the smallest percolating set for the -neighbour bootstrap process in the hypercube.

## Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question   What is the saturation number of cycles of length in the -dimensional hypercube?

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 .

Conjecture .

Keywords: coloring; hypercube

## Matchings extend to Hamiltonian cycles in hypercubes ★★

Author(s): Ruskey; Savage

Question   Does every matching of hypercube extend to a Hamiltonian cycle?

Keywords: Hamiltonian cycle; hypercube; matching

## The Crossing Number of the Hypercube ★★

Author(s): Erdos; Guy

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.

Conjecture Keywords: crossing number; hypercube 