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Theses

Graphes de Steinhaus réguliers et triangles de Steinhaus dans les groupes cycliques

Abstract : The first part of the thesis is devoted to regular Steinhaus graphs. We start by giving a new proof of a theorem due to Dymacek, which states that the Steinhaus matrix associated to an even graph is doubly symmetric, by establishing a relationship between the anti-diagonal entries of a Steinhaus matrix and the vertex degrees of its Steinhaus graph. This theorem permits us to show that a Steinhaus matrix associated to a regular graph of odd degree admits a big multi-symmetric submatrix. We then study multi-symmetric Steinhaus matrices, especially those associated to graphs which admit certain regularity. Finally, this study permits us to verify up to 1500 vertices a conjecture of Dymacek, according to which the complete graph on two vertices K2 is the only regular Steinhaus graph of odd degree, thereby improving by a factor of 12 the previous bound (117 vertices).
The second part deals with Steinhaus triangles in Z/nZ. In 1978 Molluzzo asked whether there exists, for every positive integer n≥1 and for each admissible length m, a balanced sequence of length m in Z/nZ, i.e. a sequence whose Steinhaus triangle contains each element of Z/nZ with the same multiplicity. We answer positively and completely Molluzzo's Problem in every cyclic group of order a power of 3. More generally, we construct an infinite number of balanced sequences in every finite cyclic group of odd order. This is achieved by analyzing Steinhaus triangles of arithmetic progressions in finite cyclic groups. These are the first results on this problem in Z/nZ with n> 3.
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https://tel.archives-ouvertes.fr/tel-00371329
Contributor : Jonathan Chappelon <>
Submitted on : Friday, March 27, 2009 - 2:23:18 PM
Last modification on : Tuesday, January 5, 2021 - 5:24:02 PM
Long-term archiving on: : Thursday, June 10, 2010 - 6:58:41 PM

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  • HAL Id : tel-00371329, version 1

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Jonathan Chappelon. Graphes de Steinhaus réguliers et triangles de Steinhaus dans les groupes cycliques. Mathématiques [math]. Université du Littoral Côte d'Opale, 2008. Français. ⟨tel-00371329⟩

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