Bibliographic Details
Title: |
Repeated patterns in proper colourings |
Authors: |
Conlon, David, Tyomkyn, Mykhaylo |
Publication Year: |
2020 |
Collection: |
ArXiv.org (Cornell University Library) |
Subject Terms: |
Mathematics - Combinatorics |
Description: |
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 $T$. |
Document Type: |
text |
Language: |
unknown |
Relation: |
http://arxiv.org/abs/2002.00921 |
Availability: |
http://arxiv.org/abs/2002.00921 |
Accession Number: |
edsbas.A3B7548 |
Database: |
BASE |