Edge-coloured graph homomorphisms, paths, and duality

TL;DR

This paper introduces an edge-coloured analogue of the duality theorem for directed paths and tournaments, establishing a correspondence between edge-coloured paths and graphs via homomorphisms, with simple dual constructions.

Contribution

It extends the duality theorem to edge-coloured graphs, providing explicit constructions for duals of edge-coloured paths and related graphs.

Findings

Established a duality between edge-coloured paths and graphs.

Provided explicit, simple constructions for dual graphs.

Demonstrated the applicability to edge-coloured transitive tournaments.

Abstract

We present a edge-coloured analogue of the duality theorem for transitive tournaments and directed paths. Given a edge-coloured path $P$ whose edges alternate blue and red, we construct a edge-coloured graph $D$ so that for any edge-coloured graph $G$ $$ P \to G \Leftrightarrow G \not\to D. $$ The duals are simple to construct, in particular $|V(D)|=|V(P)|-1$.