Planar digraphs of digirth four are 2-colourable

TL;DR

This paper proves that all planar directed graphs with digirth at least four can be coloured with two colours without monochromatic directed cycles, advancing understanding of graph colourability in planar digraphs.

Contribution

It establishes that planar digraphs with digirth four are 2-colourable, confirming a relaxed version of Neumann-Lara's 1985 conjecture.

Findings

Planar digraphs of digirth at least four are 2-colourable.

Supports a relaxed version of Neumann-Lara's conjecture from 1985.

Advances understanding of colourability in planar directed graphs.

Abstract

Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colourable, meaning that the vertices can be 2-coloured without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colourable.