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Regular sets in Cayley graphs

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Abstract

In a graph \(\Gamma \) with vertex set V, a subset C of V is called an (a, b)-regular set if every vertex in C has exactly a neighbors in C and every vertex in \(V\setminus C\) has exactly b neighbors in C, where a and b are nonnegative integers. In the literature, (0, 1)-regular sets are known as perfect codes and (1, 1)-regular sets are known as total perfect codes. In this paper, we prove that, for any finite group G, if a non-trivial normal subgroup H of G is a perfect code in some Cayley graph of G, then for any pair of integers a and b with \(0\leqslant a\leqslant |H|-1\) and \(0\leqslant b\leqslant |H|\) such that \(\gcd (2,|H|-1)\) divides a, H is also an (a, b)-regular set in some Cayley graph of G depending on (a, b). A similar result involving total perfect codes is also proved in the paper.

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Acknowledgements

We appreciate the four anonymous referees for their helpful comments. The first author gratefully acknowledges the financial support from China Scholarship Council (No. 201806010040). The third author was supported by the National Natural Science Foundation of China (No. 61771019) and the Research Grant Support Scheme of The University of Melbourne. The third author is grateful to Peter Cameron for introducing the concept of perfect sets to him and bringing [1] to his attention.

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Authors and Affiliations

  1. Harbin University of Science and Technology, Rongcheng Campus, Harbin, 150080, Heilongjiang, People’s Republic of China

    Yanpeng Wang

  2. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Binzhou Xia & Sanming Zhou

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  1. Yanpeng Wang
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  2. Binzhou Xia
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  3. Sanming Zhou
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Correspondence to Yanpeng Wang.

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Wang, Y., Xia, B. & Zhou, S. Regular sets in Cayley graphs. J Algebr Comb 57, 547–558 (2023). https://doi.org/10.1007/s10801-022-01181-8

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