# cycles

## 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

## 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

## Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

Conjecture   If , then contains at least cycles of length .

Keywords: chromatic number; cycles

## The Bermond-Thomassen Conjecture ★★

Author(s): Bermond; Thomassen

Conjecture   For every positive integer , every digraph with minimum out-degree at least contains disjoint cycles.

Keywords: cycles