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On a conjecture of Thomassen and Toft

✍ Scribed by Kriesell, Matthias

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 183 KB
Volume: 32
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199910)32:2<118::aid-jgt2>3.0.co;2-n

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✦ Synopsis

This article is motivated by a conjecture of Thomassen and Toft on the number s 2 (G) of separating vertex sets of cardinality 2 and the number v 2 (G) of vertices of degree 2 in a graph G belonging to the class G of all 2-connected graphs without nonseparating induced cycles. Let G denote the number of edges of the graph G. Thomassen and Toft conjectured in [C. Thomassen & B. Toft, J. Combin. Theory B 31 (1981), 199-224] the existence of a positive constant c satisfying s 2 (G) + v 2 (G) > c• G for all G ∈ G. We shall see that this is not true in general. Restricting ourselves to planar graphs, we obtain s 2 (G) + v 2 (G) > 1 5 • G for all planar G ∈ G, where 1 5 is best-possible.

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