Connected Colourings of Complete Graphs and Hypergraphs

Abstract

Gallai’s colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if we $k$-colour the edges of the complete graph, with each colour class connected, how many of the $\left( {\begin{array}{c}k\\ 3\end{array}}\right)$ triples of colours must appear as triangles? In this note we show that the ‘obvious’ conjecture, namely that there are always at least $\left( {\begin{array}{c}k-1\\ 2\end{array}}\right)$ triples, is not correct. We determine the minimum asymptotically. This answers a question of Johnson. We also give some results about the analogous problem for hypergraphs, and we make a conjecture that we believe is the ‘right’ generalisation of Gallai’s theorem to hypergraphs.