Repeated patterns in proper colourings
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 Kn 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 Kn with at most c n2 colours contains two repeats of T, while there are colourings with at most c' n3/2 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 Kn 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 ω(1) repeats of T.