Colouring Non-Even Digraphs

A colouring of a digraph as defined by Neumann-Lara in 1982 is vertex-colouring such that no monochromatic directed cycles exist. The minimal number colours required for loopless to be its dichromatic number. This quantity has been widely studied the last decades and can considered natural analogue chromatic graph. $D$ called even if every $0$-$1$-weighting edges it contains cycle total weight. We show non-even at most $2$ an optimal found polynomial time. strengthen previously known NP-hardness result showing deciding whether graph $2$-colourable remains NP-hard feedback vertex set bounded size.