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arXiv · 2605.17376

Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs

R. A. Bailey·Peter J. Cameron·Sanming Zhou
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Abstract

An equitable partition of a graph $\Ga$ is a partition $\{V_1, \ldots, V_m\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a, b)$-perfect set in $\Ga$, where $a$ is the number of neighbours in $V_1$ of each vertex in $V_1$, and $b$ is the number of neighbours in $V_1$ of each vertex in $V_2$. In this paper we first derive general necessary conditions for a regular graph to admit two equitable partitions. As a corollary we obtain necessary conditions for the existence of an $(a,b)$-perfect set in a regular graph in terms of an arbitrary equitable partition. With the help of these results we then obtain necessary conditions for the existence of an $(a,b)$-perfect set in a normal Cayley graph in terms of the irreducible characters of the underlying group.

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BibTeXRIS

R. A. Bailey, Peter J. Cameron, Sanming Zhou. 2026-05-17. Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs. https://arxiv.org/abs/2605.17376

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