Cyclic subsets in regular Dirac graphs
Nemanja Draganić, Peter Keevash, Alp Müyesser
TL;DR
The paper resolves Erdős–Faudree’s conjecture on cyclic subsets in $(n+1)$-regular graphs on $2n$ vertices by identifying the extremal structure and proving a sharp asymptotic constant: $c=\tfrac{1}{2}$ for large $n$. The authors classify Dirac graphs into bi-dense, almost two cliques, and almost bipartite, and prove that a random half-subgraph $G[\tfrac{1}{2}]$ is Hamiltonian with high probability in the first two regimes, while near-bipartite graphs determine the extremal behavior. They then establish an exact bound by analyzing the near-bipartite case and computing the exact limiting probability $p_n = \tfrac{1}{2} + \tfrac{3}{2}{\sqrt{\pi n}}^{-1} + O(n^{-3/2})$, showing the extremal graphs are the $G_n$ family formed from $K_{n-1,n+1}$ with a $2$-factor inside the larger part. The results deepen the understanding of robustness of Hamiltonicity in Dirac graphs and offer precise extremal configurations, with potential extensions to related combinatorial robustness questions like $r$-factors and Hajnal–Szemerédi-type phenomena.
Abstract
In 1996, in his last paper, Erdős 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 $1/2$.