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Clique numbers of finite unit-quadrance graphs

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Abstract

Let F be a field, and let n be a positive integer. The vertex set of the unit-quadrance graph UD(F, n) is the set of all n-tuples with entries in F; two vertices are adjacent if and only if the sum of the squares of the entries in their difference is 1 (this generalizes Euclidean distance). These graphs have previously been studied by Vinh as well as by Medrano, Myers, Stark, and Terras. In this paper, we completely determine the clique numbers of UD(F, n) for all finite fields F and for all n. Some other properties of these graphs, such as bipartiteness, connectedness, and conditions under which the chromatic number strictly exceeds the clique number, are briefly discussed.

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Acknowledgements

The author wishes to thank Carlos Augusto Di Prisco De Venanzi for directing him to an appropriate reference for the independence of (AC2) from (ZF), as well as Ramin Takloo-Bighash for his assistance with Sect. 5.5.2. We also thank Craig Timmons for pointing out some articles in which these graphs had previously been studied. Moreover, we thank both reviewers for their excellent suggestions, which greatly improved and in many cases substantially streamlined this article.

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  1. Department of Mathematics, California State University - Los Angeles, 5151 State University Drive, Los Angeles, CA, 90032, USA

    Mike Krebs

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  1. Mike Krebs
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Correspondence to Mike Krebs.

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Krebs, M. Clique numbers of finite unit-quadrance graphs. J Algebr Comb 57, 1–20 (2023). https://doi.org/10.1007/s10801-022-01157-8

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  • DOI: https://doi.org/10.1007/s10801-022-01157-8

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