Rainbow and orthogonal paths in factorizations of Kn

András Gyárfás, Mehdi Mhalla

Research output: Contribution to journalArticle

9 Citations (Scopus)


For n even, a factorization of a complete graph Kn is a partition of the edges into n-1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizations have orthogonal paths. Assisted by a simple edge-switching algorithm, here we show that for n≥8, the rotational factorization of Kn, GKn has orthogonal paths. We prove that this algorithm finds a rainbow path with at least (2n+1)/3 vertices in any factorization of Kn (in fact, in any proper coloring of K n). We also give some problems and conjectures about the properties of the algorithm.

Original languageEnglish
Pages (from-to)167-176
Number of pages10
JournalJournal of Combinatorial Designs
Issue number3
Publication statusPublished - May 2010


  • Factorizations of complete graphs
  • Rainbow and orthogonal paths

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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