Decomposing edge-coloured complete symmetric digraphs into monochromatic paths

TL;DR

This paper proves a conjecture about decomposing edge-coloured complete symmetric digraphs into monochromatic paths, extending previous results to cover specific path length restrictions across multiple colours.

Contribution

It introduces a new decomposition theorem for countable edge-coloured complete symmetric digraphs with path length constraints, generalizing prior conjectures.

Findings

Every such digraph can be covered by the product of path length bounds plus one monochromatic paths.

The result confirms a conjecture by Guggiari for countable cases.

The decomposition applies to graphs with no directed paths of specified lengths in certain colours.

Abstract

Confirming and extending a conjecture by Guggiari, we show that every countable $(r+1)$-edge-coloured complete symmetric digraph containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic directed paths in colour $r+1$.