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arXiv · 2503.01826

Cyclic subsets in regular Dirac graphs

Nemanja Draganić·Peter Keevash·Alp Müyesser
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Abstract

In 1996, in his last paper, Erd\H{o}s asked the following question that he formulated together with Faudree: is there a positive $c$ such that any $(n+1)$-regular graph $G$ on $2n$ vertices contains at least $c 2^{2n}$ distinct vertex-subsets $S$ that are cyclic, meaning that there is a cycle in $G$ using precisely the vertices in $S$. We answer this question in the affirmative in a strong form by proving the following exact result: if $n$ is sufficiently large and $G$ minimises the number of cyclic subsets then $G$ is obtained from the complete bipartite graph $K_{n-1,n+1}$ by adding a $2$-factor (a spanning collection of vertex-disjoint cycles) within the part of size $n+1$. In particular, for $n$ large, this implies that the optimal $c$ in the problem is precisely $1/2$.

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Nemanja Draganić, Peter Keevash, Alp Müyesser. 2025-03-03. Cyclic subsets in regular Dirac graphs. https://arxiv.org/abs/2503.01826

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