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Three-Dimensional Graph Products with Unbounded Stack-Number

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Abstract

We prove that the stack-number of the strong product of three n-vertex paths is \(\Theta (n^{1/3})\). The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest \(\Delta _0\) such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree \(\Delta _0\). We show that \(\Delta _0\in \{6,7\}\).

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Notes

  1. All graphs in this paper are simple and, unless explicitly stated otherwise, undirected and finite. Let V(G) and E(G) denote the vertex-set and edge-set of a graph G. Let \(\mathbb {N}=\{1,2,\dots \}\) and \(\mathbb {N}_0=\mathbb {N}\cup \{0\}\).

  2. A key step in the proof of (1.1) is the application of the Hex Lemma, which holds for any triangulation of \(P_n\square P_n\) (see [61] for example). Theorem 1.3 then follows by the method in [22].

  3. See [11, Sect. 12] for background on simplicial complexes.

  4. This definition is inspired by the definition of a bramble in a graph, which is used in graph minor theory. A bramble of a graph is only required to satisfy the first and second among the conditions we impose.

  5. A topological space X is path-connected if there is a path joining any two points of X; that is, for all \(x,y \in X\) there exists a continuous map \(f:[0,1] \rightarrow X\) such that \(f(0)=x\) and \(f(1)=y\).

  6. A topological space is simply connected if it is path-connected and any two paths with the same endpoints can be continuously transformed into each other.

  7. See [39, Thm. 1.20] for the general statement of van Kampen’s theorem, or [18, Sect. 2.1.3] for an example of the application of the theorem in the setting generalizing Lemma 3.4.

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Acknowledgements

This work was initiated at the Workshop on Graph Product Structure Theory (BIRS21w5235) at the Banff International Research Station, 21–26 November 2021. Thanks to the other organisers and participants. Thanks to both referees for many helpful suggestions.

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Authors and Affiliations

  1. Department of Computer Science, University of California, Irvine, CA, USA

    David Eppstein

  2. School of Mathematics, Monash University, Melbourne, Australia

    Robert Hickingbotham & David R. Wood

  3. Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Laura Merker

  4. Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada

    Sergey Norin

  5. Department of Theoretical Computer Science, Jagiellonian University, Kraków, Poland

    Michał T. Seweryn

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  1. David Eppstein
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Norin is supported by an NSERC Discovery Grant. Hickingbotham is supported by an Australian Government Research Training Program Scholarship. Wood is supported by the Australian Research Council. Eppstein is supported in part by NSF grant CCF-2212129.

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Eppstein, D., Hickingbotham, R., Merker, L. et al. Three-Dimensional Graph Products with Unbounded Stack-Number. Discrete Comput Geom 71, 1210–1237 (2024). https://doi.org/10.1007/s00454-022-00478-6

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