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arXiv · 2609.09414

Regular sets of circulant quartic graphs

A. Abdollahi·J. Bagherian·F. Jafari·M. Khatami·Z. Shokoohi·R. Sobhani
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Abstract

For a graph $\Gamma=(V,E)$ and nonnegative integers $a$ and $b$, a nonempty proper subset $C \subset V$ 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 $C$. In this paper, we study the existence of such sets in connected Cayley graph $\Gamma = \operatorname{Cay}(\mathbb{Z}_n, S)$. We establish a necessary and sufficient condition for the existence of $(0, |S|)$-regular sets and identify additional conditions under which no such set can exist. We further prove that $(|S|, 0)$-regular sets do not occur in $\Gamma$, and more generally, that no connected Cayley graph $\operatorname{Cay}(G,S)$ contains a $(1, |S|)$-regular set. As a main result, we determine the existence and nonexistence of $(a,b)$-regular sets in connected circulant quartic graphs for all possible values of $a$ and $b$.

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A. Abdollahi, J. Bagherian, F. Jafari, M. Khatami, Z. Shokoohi, R. Sobhani. 2026-09-08. Regular sets of circulant quartic graphs. https://arxiv.org/abs/2609.09414

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