# hypercube

## Weak saturation of the cube in the clique ★

**Problem**

Determine .

Keywords: bootstrap percolation; hypercube; Weak saturation

## 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 ★★

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

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

## 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**.

## 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.

**Conjecture**

Keywords: crossing number; hypercube