Multicoloured Hamilton Cycles

Mi hael Albert, Alan Frieze and Bru e Reed Department of Mathemati s, Carnegie-Mellon University, Pittsburgh, U.S.A.y Submitted: April 25th,1995 A epted May 9th, 1995 Abstra t The edges of the omplete graph Kn are oloured so that no olour appears more than d ne times, where < 1=32 is a onstant. We show that if n is suÆ iently large then there is a Hamiltonian y le in whi h ea h edge is a di erent olour, thereby proving a 1986 onje ture of Hahn and Thomassen [9℄. We prove a similar result for the omplete digraph with < 1=64. We also show, by essentially the same te hnique, that if t 3, < (2t2(1 + t)) 1, no olour appears more than d ne times and tjn then the verti es an be partitioned into n=t t sets K1;K2; : : : ;Kn=t su h that the olours of the n(t 1)=2 edges ontained in the Ki's are distin t. The proof te hnique follows the lines of Erd} os and Spen er's [4℄ modi ation of the Lo al Lemma [1℄. Current address of Bru e Reed: Equipe Combinatoire, CNRS, Universit e Pierre et Marie Curie, 4 Pla e Jussieu, Paris, Fran e yAlan Frieze: partially supported by NSF grant CCR-9225008