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Repeated patterns in proper colourings

Conlon, David and Tyomkyn, Mykhaylo (2020) Repeated patterns in proper colourings. . (Unpublished)

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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 cn² colours contains two repeats of T, while there are colourings with at least 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 cn 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.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Conlon, David0000-0001-5899-1829
Record Number:CaltechAUTHORS:20200914-134941914
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105380
Deposited By: Tony Diaz
Deposited On:14 Sep 2020 21:54
Last Modified:14 Sep 2020 21:54

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