# Cycles through ten vertices in 3-connected cubic graphs

## Abstract

It is known that there exists a cycle through any nine vertices of a 3-connected cubic graphG. Here we show that if an edge is removed from such a graph, then there is still a cycle through any five vertices. Furthermore, we characterise the circumstances in which there fails to be a cycle through six. As corollaries we are able to prove that a 3-connected cubic graph has a cycle through any specified five vertices and one edge, and to classify the conditions under which it has a cycle through four chosen vertices and two edges.

We are able to use the five and six vertex results to show that a 3-connected cubic graph has a cycle which passes through any ten given vertices if and only if the graph is not contractible to the Petersen graph in such a way that the ten vertices each map to a distinct vertex of the Petersen graph.

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## References

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J. A. Bondy andU. S. R. Murty,Graph Theory with Applications, Macmillan, London, 1976.

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M. N. Ellingham, Constructing certain cubic graphs,Combinatorial Mathematics IX, Lecture Notes in Maths., No.952, Springer, Berlin, 1982, 252–274.

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A. K. Kelmans andM. V. Lomonosov, A cubic 3-connected graph having no cycles through given 10 vertices has the “Petersen” form,Amer. Math. Soc. Abstracts, No. 82T-05-260,3 (1982), 283.

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Reprints and Permissions

Ellingham, M.N., Holton, D.A. & Little, C.H.C. Cycles through ten vertices in 3-connected cubic graphs. Combinatorica 4, 265–273 (1984). https://doi.org/10.1007/BF02579136

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• 05 C 40
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