Well-spread sequences and edge-labellings with constant Hamilton-weight

A sequence (a i ) of integers is well-spread if the sums a i +a j , for i<j , are all different. For a fixed positive integer r , let W r (N) denote the maximum integer n for which there exists a well-spread sequence 0≤ a 1 <…<a n ≤ N with a i ≡ a j (b mod r) for all i , j . We give a new proof that W r (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 K n with the property that every Hamilton cycle has the same length; we prove that 2n 2 -O(n 3/2 )<Λ(n)<2n 2 +O(n 61/40 ) .

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