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Cycles and Girth in Pebble Assignment Graphs

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Abstract

An assignment graph \([S_G]\) is a single-rooted Hasse diagram depicting all possible states resulting from a prescribed pebble assignment \(S_G\) on a simple graph G. In this paper, we construct assignment graphs of every possible (even) girth and give necessary and sufficient conditions for \([S_G]\) to have girth 4. We extend the notion of an assignment graph to that of a multiassignment graph (a multirooted Hasse diagram formed by amalgamating two or more assignment graphs on G) and resolve the question: When can a multiassignment graph be a subgraph of some assignment graph? Resolution of this question is critical to our main result: Every possible cycle type of girth at most 2n can be simultaneously realized in a suitable assignment graph. The paper concludes with a proof that the girth of \([S_G]\) is limited to \(4,6,\infty \) when G is a forest.

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Acknowledgements

The authors wish to thank two referees for their insightful comments that contributed heavily to the quality of this paper.

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The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Dimacs–Rutgers University, Piscataway, NJ, 08854, USA

    E. Fiorini

  2. Villanova University, Villanova, PA, 19085, USA

    G. Johnston & A. Woldar

  3. University Scholars Program, West Chester, PA, 19380, USA

    M. Lind

  4. Kutztown University of Pennsylvania, Kutztown, PA, 19530, USA

    T. W. H. Wong

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  1. E. Fiorini
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  4. A. Woldar
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  5. T. W. H. Wong
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Contributions

Authors contributed to the study conception and design. Material preparation and analysis were performed by Max Lind, Eugene Fiorini and Andrew Woldar. The first draft of the manuscript was written by Andrew Woldar and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to E. Fiorini.

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Appendices

Appendix A. Construct of a girth 8 assignment graph

figure a

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Fiorini, E., Johnston, G., Lind, M. et al. Cycles and Girth in Pebble Assignment Graphs. Graphs and Combinatorics 38, 154 (2022). https://doi.org/10.1007/s00373-022-02552-5

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  • DOI: https://doi.org/10.1007/s00373-022-02552-5

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