Monochromatic paths in $2$-edge coloured graphs and hypergraphs

TL;DR

This paper proves that complete 3-uniform hypergraphs can be partitioned into two monochromatic tight paths of different colours, and extends results to bipartite graphs and r-uniform hypergraphs, advancing understanding of monochromatic path partitions.

Contribution

It resolves a 2013 question by showing a partition into two monochromatic tight paths in 3-uniform hypergraphs and provides bounds for r-uniform hypergraphs and bipartite graphs.

Findings

Complete 3-uniform hypergraphs can be partitioned into two monochromatic tight paths.

A lower bound is given for the number of paths needed in r-uniform hypergraphs.

Complete bipartite graphs can be partitioned into a monochromatic cycle and path, except in split colourings.

Abstract

We answer a question of Gy\'arf\'as and S\'ark\"ozy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete r-partite r-uniform hypergraph. Finally, we show that any 2-edge-coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.