Repeated patterns in proper colourings

TL;DR

This paper investigates the minimum number of colours needed in proper edge-colourings of complete graphs to avoid multiple vertex-disjoint repeats of a fixed graph, using combinatorial, probabilistic, and algebraic methods.

Contribution

It introduces bounds and constructions for avoiding repeated subgraphs in edge-colourings, extending understanding of pattern repetition in graph colourings.

Findings

Proper colourings with linear in n colours can contain multiple repeats of trees.

Existence of colourings with sublinear in n^{(m+1)/m} colours avoiding repeats of trees.

For graphs with cycles, bounded repeats are achievable with linear in n colours.

Abstract

For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We study this function and its generalisation to more than two copies using a variety of combinatorial, probabilistic and algebraic techniques. For example, we show that for any tree $T$ there exists a constant $c$ such that any proper edge-colouring of $K_n$ with at most $c n^2$ colours contains two repeats of $T$, while there are colourings with at most $c' n^{3/2}$ colours for some absolute constant $c'$ containing no three repeats of any tree with at least two edges. We also show that for any graph $H$ containing a cycle there exist $k$ and $c$ such that there is a proper edge-colouring of $K_n$ with at most $c n$ colours containing no $k$ repeats of $H$, while, for a tree $T$ with $m$ edges, a colouring with $o(n^{(m+1)/m})$ colours contains $\omega(1)$ repeats of $T$.