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On a question of van Aardt et al. on destroying all longest cycles. (English) Zbl 1409.05119
Summary: We describe an infinite family of 2-connected graphs, each of which has the property that the intersection of all longest cycles is empty. In particular, we present such graphs with circumference 10, 13, and 16. This settles a question of S. A. van Aardt et al. [Discrete Appl. Math. 186, 251–259 (2015; Zbl 1311.05094)] concerning the existence of such graphs for all but one case, namely circumference 11. We also present a 2-connected graph of circumference 11 in which all but one vertex are avoided by some longest cycle.
MSC:
05C38 Paths and cycles
05C12 Distance in graphs
05C35 Extremal problems in graph theory
05C40 Connectivity
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