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Distinct Sums Modulo n and Tree Embeddings

Published online by Cambridge University Press:  14 February 2002

ANDRÉ E. KÉZDY  and
HUNTER S. SNEVILY
ANDRÉ E. KÉZDY
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (e-mail: kezdy@louisville.edu)
HUNTER S. SNEVILY
Affiliation:
Department of Mathematics, University of Idaho, Moscow, ID 83844, USA (e-mail: snevily@uidaho.edu)
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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 [les ] 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 [les ] 32(2r+4)n2+1.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 1 , January 2002 , pp. 35 - 42
Copyright
2002 Cambridge University Press

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