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All My Favorite Conjectures Are Critical

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Graph Theory

Part of the book series: Problem Books in Mathematics ((PBM))

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Abstract

My favorite graph theory conjectures involve the effects of edge removal on the diameter of a graph and the effects of edge addition on the domination and total domination numbers of a graph. Loosely speaking, “criticality” means that the value of the parameter in question always changes under the graph modification. This chapter presents five conjectures concerning criticality, namely, a conjecture by Sumner and Blitch on the criticality of domination upon edge addition, a conjecture by Murty and Simon on the criticality of diameter upon edge removal, and three conjectures on the criticality of total domination upon edge addition. These last three conjectures involving total domination are closely related, and surprisingly, a solution to one of them would provide a solution to the Murty-Simon Conjecture on diameter.

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Acknowledgements

First I thank the editors for their initial conception of this volume and their subsequent hard work to complete it. I also want to thank Wyatt Desormeaux and the anonymous reviewers of this chapter for their many helpful comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, East Tennessee State University, Johnson City, 37614, TN, USA

    Teresa W. Haynes

  2. Department of Mathematics, University of Johannesburg, Auckland Park, Johannesburg, South Africa

    Teresa W. Haynes

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  1. Teresa W. Haynes
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Correspondence to Teresa W. Haynes .

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Editors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, California, USA

    Ralucca Gera

  2. School of Computing, Clemson University, Clemson, South Carolina, USA

    Stephen Hedetniemi

  3. Department of Mathematics, Virginia Commonwealth University, Richmond, Virginia, USA

    Craig Larson

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Haynes, T.W. (2016). All My Favorite Conjectures Are Critical. In: Gera, R., Hedetniemi, S., Larson, C. (eds) Graph Theory. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31940-7_5

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