Edge-coloured graph homomorphisms, paths, and duality

Abstract

We present a 2-edge-coloured analogue of the duality theorem for transitive tournaments and directed paths. Given a 2-edge-coloured path $P$ whose edges alternate blue and red, we construct a 2-edge-coloured graph $D$ so that for any 2-edge-coloured graph $G$,

$$P\to G⟺G\nrightarrow D.$$

The duals are simple to construct, in particular $|V(D)|=|V(P)|-1.$