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Contractible Non-edges in 3-Connected Infinite Graphs

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Abstract

In this paper, we prove that for any 3-connected finite graph of order n\((n\ge 6)\), the number of contractible non-edges is at most \(\frac{n(n-5)}{2}\). All the extremal graphs with at least seven vertices are characterized to be 4-connected 4-regular. By generalizing a result of Kriesell (J Comb Theory Ser B 74:192–201, 1998), we also characterize all 3-connected graphs (finite or infinite) that does not contain any contractible non-edges. In particular, every non-complete 3-connected infinite graph contains a contractible non-edge.

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Acknowledgements

The author would like to thank the referees for pointing out errors and giving suggestions that greatly improve the accuracy and presentation of the paper.

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  1. Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Tsz Lung Chan

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  1. Tsz Lung Chan
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Correspondence to Tsz Lung Chan.

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Chan, T.L. Contractible Non-edges in 3-Connected Infinite Graphs. Graphs and Combinatorics 35, 1447–1458 (2019). https://doi.org/10.1007/s00373-019-02091-6

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  • DOI: https://doi.org/10.1007/s00373-019-02091-6

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