# acyclic

## Non-edges vs. feedback edge sets in digraphs ★★★

Author(s): Chudnovsky; Seymour; Sullivan

For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.

Conjecture  If is a simple digraph without directed cycles of length , then .

Keywords: acyclic; digraph; feedback edge set; triangle free

## The Two Color Conjecture ★★

Author(s): Neumann-Lara

Conjecture   If is an orientation of a simple planar graph, then there is a partition of into so that the graph induced by is acyclic for .

Keywords: acyclic; digraph; planar