David Conlon, Mykhaylo Tyomkyn

Research output: Contribution to journalArticlepeer-review


For a fixed graph H, what is the smallest number of colors C such that there is a proper edge-coloring of the complete graph Kn with C colors containing no two vertex-disjoint color-isomorphic copies, or repeats, of H? We study this function and its generalization 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-coloring of Kn with at most cn2 colors contains two repeats of T, while there are colorings with at most c\prime n3/2 colors for some absolute constant c\prime 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-coloring of Kn with at most cn colors containing no k repeats of H, while for a tree T with m edges, a coloring with o(n(m+1)/m) colors contains \omega (1) repeats of T.

Original languageEnglish
Pages (from-to)2249-2264
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - 2021
Externally publishedYes


  • Coloring
  • Extremal problems
  • Graphs


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