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The Toughness of a Toroidal Graph with Connectivity 3 and Girth 6 is Exactly 1

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Abstract.

We show that every toroidal graph with connectivity 3 and girth 6 is bipartite. This implies that its toughness is at most 1. This answers a question in Goddard, Plummer, and Swart [3] in which it was shown that such a graph has toughness at least 1.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

    Weiting Cao

  2. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

    Michael J. Pelsmajer

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  1. Weiting Cao
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  2. Michael J. Pelsmajer
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Correspondence to Michael J. Pelsmajer.

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Acknowledgments We would like to thank Douglas B. West for helpful comments.

Final version received: November 17, 2003

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Cao, W., Pelsmajer, M. The Toughness of a Toroidal Graph with Connectivity 3 and Girth 6 is Exactly 1. Graphs and Combinatorics 20, 181–183 (2004). https://doi.org/10.1007/s00373-004-0558-z

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  • DOI: https://doi.org/10.1007/s00373-004-0558-z

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