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Graceful Trees: Statistics and Algorithms


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Horton, M 2003 , 'Graceful Trees: Statistics and Algorithms', Honours thesis, University of Tasmania.

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The Graceful Tree Conjecture is a problem in graph theory that dates back to 1967. It suggests that every tree on n nodes can be labelled with the integers [1..n] such that the edges, when labelled with the difference between their endpoint node labels, are uniquely labelled with the integers [1..n-1]. To date, no proof or disproof of the conjecture has been found, but all trees with up to 28 vertices have been shown to be graceful. The conjecture also leads to a problem in algorithm design for efficiently finding graceful labellings for trees. In this thesis, a new graceful labelling algorithm is described and used to show that all trees on 29 vertices are graceful. A study is also made of statistical trends in the proportion of tree labellings that are graceful. These trends offer strong additional evidence that every tree is graceful.

Item Type: Thesis - Honours
Authors/Creators:Horton, M
Keywords: Graceful Trees, Ringel, algorithm, graceful labelling, graceful tree conjecture
Publisher: Honours thesis, University of Tasmania
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