Distinct Sums Modulo n and Tree Embeddings

  • Authors:

  • André E. Kézdy;Hunter S. Snevily

  • Affiliations:

  • Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (e-mail: kezdy@louisville.edu);Department of Mathematics, University of Idaho, Moscow, ID 83844, USA (e-mail: snevily@uidaho.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2002

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Abstract

In this paper we are concerned with the following conjecture.Conjecture. For any positive integers n and k satisfying k n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.We prove this conjecture when 2k ≤ n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t ≤ 32(2r+4)n2+1.