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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 23(2) (2003) 287-307
DOI: 10.7151/dmgt.1203


Scott Jones

Department of Mathematical Sciences
University of Montana
Missoula MT 59812-0864, USA

P. Mark Kayll1

Department of Mathematical Sciences
University of Montana
Missoula MT 59812-0864, USA

Bojan Mohar2

Department of Mathematics
University of Ljubljana
Jadranska 19
1000 Ljubljana, Slovenia

Walter D. Wallis

Department of Mathematics
Southern Illinois University
Carbondale IL 62901-4408, USA


Is it possible to label the edges of Kn with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question's perhaps surprising affirmative answer. More generally, we address the question that arises when ``Hamilton cycle'' is replaced by ``k-factor'' for nonnegative integers k. Such edge-labellings are in correspondence with certain vertex-labellings, and the link allows us to determine (up to a constant factor) the growth rate of the maximum edge-label in a ``most efficient'' injective metric trivial-TSP labelling.

Keywords: graph labelling, complete graph, travelling salesman problem, Hamilton cycle, one-factor, two-factor, k-factor, constant-weight, local matching conditions, edge label growth-rate, Sidon sequence, well-spread sequence.

2000 Mathematics Subject Classification: Primary 05C78; Secondary 05C70, 11B75, 90C27.


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Received 11 December 2001
Revised 7 May 2002


1Contact author; on leave at University of Ljubljana-the author thanks the Department of Mathematics and the Institute of Mathematics, Physics and Mechanics for their hospitality.
2Supported in part by the Ministry of Science and Technology of Slovenia, Research Program P1-0507-0101.