Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

For an increasing monotone graph property $\mP$ the \emph{local resilience} of a graph $G$ with respect to $\mP$ is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of the edges of $H$ from $G$ creates a graph that does not possesses $\mP$. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model $\GNP$ and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random $d$-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive $\epsilon$ and large enough values of $d$ with high probability the local resilience of the random $d$-regular graph, $\GND$, with respect to being Hamiltonian is at least $(1-\epsilon)d/6$. We also prove that for the Binomial random graph model $\GNP$, for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-\epsilon)np/6$. Finally, we apply similar techniques to Positional Games and prove that if $d$ is large enough then with high probability a typical random $d$-regular graph $G$ is such that in the unbiased Maker-Breaker game played on the edges of $G$, Maker has a winning strategy to create a Hamilton cycle.

💡 Research Summary

The paper investigates the concept of local resilience – the smallest integer r such that one can delete a subgraph of maximum degree r and destroy a given monotone graph property 𝒫 – in the setting of random regular graphs. While previous work focused mainly on the Erdős–Rényi model 𝔾(n,p) and on pseudo‑random graphs, the authors turn to the random d‑regular graph 𝔾ₙ,𝑑 (with constant degree) and to the binomial model in the regime p≫log n/n. Their main results can be grouped into three themes: (i) resilience of structural properties (Hamiltonicity, perfect matching, vertex‑ and edge‑connectivity) in 𝔾ₙ,𝑑, (ii) analogous resilience bounds for 𝔾(n,p), and (iii) applications to unbiased Maker–Breaker positional games played on the edge set of a random regular graph.

Technical framework. The authors start by formalising local resilience for a monotone property 𝒫. They then recall that a random d‑regular graph, generated via the configuration model, exhibits strong expansion: for every set S of size at most αn (α a small constant depending on d), the neighbourhood satisfies |N(S)|≥(1+β)|S| with high probability, where β>0 is a function of d. This expansion is equivalent to a spectral gap λ(G)≤2√(d−1)+o(1). Using these facts, the paper leverages classic tools such as Dirac’s theorem, Hall’s condition, and Pósa’s rotation‑extension technique.

Resilience of Hamiltonicity. For any fixed ε>0 and sufficiently large d, the authors prove that with high probability the local resilience of 𝔾ₙ,𝑑 with respect to being Hamiltonian is at least (1−ε)d/6. The proof proceeds by assuming an adversarial subgraph H of maximum degree r<(1−ε)d/6 is removed. The remaining graph G′ has minimum degree at least d−r≥(5+ε)d/6. By the expansion property, every small vertex set expands by a factor >1, guaranteeing that G′ satisfies the conditions required for Pósa’s rotation‑extension method. Consequently G′ contains a Hamilton cycle. The same line of reasoning yields a resilience bound of (1−ε)np/6 for 𝔾(n,p) when p>K log n/n for a sufficiently large constant K.

Other properties. For perfect matchings the authors invoke Hall’s theorem: the expansion ensures that for any set S, |N(S)|≥|S|, so a matching cannot be destroyed as long as the deleted subgraph has degree below the same (1−ε)d/6 threshold. Vertex‑ and edge‑connectivity follow from the observation that a graph with minimum degree at least k is automatically k‑connected; thus the same resilience bound guarantees preservation of any prescribed connectivity level up to (1−ε)d/6.

Maker–Breaker game. In the unbiased game each player claims one free edge per turn. The authors design a strategy for Maker that maintains the invariant that the subgraph he has built never exceeds the resilience threshold r<(1−ε)d/6. At each move Maker selects an edge that either (a) connects a low‑degree vertex to preserve expansion, or (b) closes a long path using the rotation‑extension technique. Because the underlying random regular host graph retains strong expansion even after Breaker’s deletions, Maker can repeatedly apply Pósa’s method to extend his path until it becomes a Hamilton cycle. The analysis shows that for sufficiently large d, Maker wins with probability tending to one, regardless of Breaker’s optimal play.

Implications and outlook. The work demonstrates that random regular graphs, despite having a fixed degree, are as robust as dense Erdős–Rényi graphs with respect to several fundamental properties. The combination of spectral methods, probabilistic expansion, and combinatorial game theory provides a versatile toolkit that can be adapted to other regular or pseudo‑regular models, and suggests that networks designed with uniform degree constraints can still achieve high fault‑tolerance and strategic resilience.