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Exact Distance Graphs of Product Graphs

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Abstract

Given a graph G, the exact distance-p graph \(G^{[\natural p]}\) has V(G) as its vertex set, and two vertices are adjacent whenever the distance between them in G equals p. We present formulas describing the structure of exact distance-p graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-2 graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-p graphs of hypercubes \(Q_n\) are also studied. As these graphs contain generalized Johnson graphs as induced subgraphs, we use some known constructions of their colorings. These constructions are applied for colorings of the exact distance-p graphs of hypercubes with the focus on the chromatic number of \(Q_{n}^{[\natural p]}\) for \(p\in \{n-2,n-3,n-4\}\).

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Acknowledgements

We thank the reviewers for their very careful reading of the paper. This work was performed with the financial support of the bilateral project “Distance-constrained and game colorings of graph products” (BI-FR/18-19-Proteus-011). B. B. and S. K. acknowledge the financial support from the Slovenian Research Agency, Javna Agencija za Raziskovalno Dejavnost RS (research core funding no. P1-0297, project Contemporary invariants in graphs no. J1-9109, and project J1-1693).

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Boštjan Brešar & Sandi Klavžar

  2. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Boštjan Brešar & Sandi Klavžar

  3. Laboratoire LIB, Université de Bourgogne Franche-Comté, Dijon, France

    Nicolas Gastineau & Olivier Togni

  4. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

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  1. Boštjan Brešar
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  2. Nicolas Gastineau
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  3. Sandi Klavžar
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  4. Olivier Togni
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Correspondence to Olivier Togni.

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Brešar, B., Gastineau, N., Klavžar, S. et al. Exact Distance Graphs of Product Graphs. Graphs and Combinatorics 35, 1555–1569 (2019). https://doi.org/10.1007/s00373-019-02089-0

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  • DOI: https://doi.org/10.1007/s00373-019-02089-0

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