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An excluded minor theorem for the octahedron

✍ Scribed by Maharry, John


Publisher
John Wiley and Sons
Year
1999
Tongue
English
Weight
221 KB
Volume
31
Category
Article
ISSN
0364-9024

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✦ Synopsis


In this article it is shown that every 4-connected graph that does not contain a minor isomorphic to the octahedron is isomorphic to the square of an odd cycle.


πŸ“œ SIMILAR VOLUMES


An excluded minor theorem for the Octahe
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## Abstract Let __G__ be the unique 4‐connected simple graph obtained by adding an edge to the Octahedron. Every 4‐connected graph that does not contain a minor isomorphic to __G__ is either planar or the square of an odd cycle. Β© 2007 Wiley Periodicals, Inc. J Graph Theory 57: 124–130, 2008

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This paper strengthens the excluded-minor characterization of GF(4)-representable matroids. In particular, it is shown that there are only finitely many 3-connected matroids that are not GF(4)-representable and that have no U 2, 6 -, U 4, 6 -, P 6 -, F & 7 -, or (F & 7 )\*-minors. Explicitly, these

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## Abstract In 1890, Heawood established the upper bound $H ( \varepsilon )= \left \lfloor 7+\sqrt {24\varepsilon +1}/{2}\right \rfloor$ on the chromatic number of every graph embedded on a surface of Euler genus Ξ΅ β‰₯ 1. Almost 80 years later, the bound was shown to be tight by Ringel and Youngs. Th