Discrete Mathematics & Theoretical Computer Science
| author: | Kayll, P. Mark |
|---|---|
| title: | Well-spread sequences and edge-labellings with constant Hamilton-weight |
| keywords: | Well-spread, weak Sidon, graph labelling, Hamilton cycle |
| abstract: | A sequence (ai) of integers is well-spread if the sums ai+aj, for i<j, are all different. For a fixed positive integer r,
let Wr(N) denote the maximum integer n for which there exists a
well-spread sequence 0≤ a1<…<an≤ N with
ai≡ aj(b mod r) for all i, j.
We give a new proof that Wr(N)<(N/r)1/2+O((N/r)1/4); our
approach improves a bound of Ruzsa [Acta.Arith. 65
(1993), 259--283] by decreasing the
implicit constant, essentially from 4 to √3.
We apply this
result to verify a conjecture of Jones et al. from
[Discuss. Math. Graph Theory 23 (2003),
287--307]. The application concerns the
growth-rate of the maximum label Λ(n) in a `most-efficient'
metric, injective edge-labelling of Kn with the property that every
Hamilton cycle has the same length; we prove that
2n2-O(n3/2)<Λ(n)<2n2+O(n61/40).
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| reference: | Kayll, P. Mark (2004), Well-spread sequences and edge-labellings with constant Hamilton-weight, Discrete Mathematics and Theoretical Computer Science 6, pp. 401-408 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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