--- title: Hamiltonian Resilience in Graph Theory url: https://www.emergentmind.com/topics/hamiltonian-resilience type: topic --- # Hamiltonian Resilience in Graph Theory Searching arXiv for recent and foundational papers on Hamiltonian resilience to ground the article. Hamiltonian resilience is the study of how Hamiltonian structure persists under perturbation. In graph theory, the term usually denotes robustness of Hamiltonicity, Hamiltonian connectedness, or related spanning-cycle and spanning-path properties under adversarial deletions constrained locally, globally, by degree sequence, by motif counts, or by mixed vertex/edge faults. Across random graphs, random regular graphs, pseudorandom graphs, directed graphs, hypergraphs, geometric graphs, powers of cycles, and interconnection networks, the recurring question is whether a host that is Hamiltonian in an existential sense remains Hamiltonian after substantial damage, and what the correct threshold of damage is [1710.00799], [1903.05052], [2507.22807], [1111.5391]. ## 1. Formal notions and scope Several non-equivalent resilience notions are used in the literature, and the choice of notion is part of the subject rather than a technicality. In the standard undirected local form, a graph \(G\) is \(\alpha\)-resilient with respect to a property \(\mathcal P\) if every spanning subgraph \(H\subseteq G\) satisfying \[ \deg_H(v)\le \alpha\,\deg_G(v)\qquad\text{for every }v\in V(G) \] leaves \(G-H\) with property \(\mathcal P\) [1710.00799]. In \(d\)-regular hosts, this is equivalent to asking whether every spanning subgraph with minimum degree at least \((1/2+\gamma)d\) is Hamiltonian, which is the Dirac-type formulation used for pseudorandom and random regular graphs [2507.22807], [1903.05052]. For digraphs, the local deletion budget is split into in- and out-degrees. The directed local resilience of a digraph \(D\) with respect to directed Hamiltonicity is \[ r_{\ell}(D, {\mathcal H}) = \min \{r : \exists H \subseteq D \textrm{ such that } \Delta^+(H) \leq r, \Delta^-(H) \leq r \textrm{ and } D \setminus H \notin {\mathcal H}\}, \] so robustness is measured simultaneously against incoming and outgoing deletions [1404.4734]. A different notion is global resilience, \[ r_g(G,\mathcal H)=\min\{\,|E(H)|: H\subseteq G,\; G\setminus H \text{ is not Hamiltonian}\,\}, \] which controls only the total number of removed edges rather than a per-vertex budget [2206.15235]. Other refinements include \(\mathbf{k}\)-resilience, where the deletion budget varies by vertex, and degree-sequence resilience, where the deleted subgraph must satisfy Pósa- or Chvátal-type degree conditions [1101.3099], [1810.12433]. A further generalization is \(H\)-resilience. For a fixed graph \(H\), \(r_H(G,\mathcal P)\) measures how many copies of \(H\) incident to each vertex may be destroyed while preserving \(\mathcal P\). For the square of a Hamilton cycle, triangle-resilience with \(H=K_3\) is the relevant notion, because ordinary edge-based local resilience is too weak to protect a structure whose local building blocks are triangles [1809.07534]. In fault-tolerant interconnection networks, Hamiltonian resilience is often endpoint-specific rather than cycle-centric. For an \(n\)-dimensional twisted hypercube-like network \(G\), faults may be vertices, edges, or both, \[ F\subseteq V(G)\cup E(G), \] and the question is whether \(G-F\) still contains a Hamiltonian or near-Hamiltonian path between prescribed terminals \(s,t\) [1111.5391]. | Notion | Constraint on deletions | Typical target | |---|---|---| | Local resilience | Per-vertex degree budget | Hamilton cycle | | Directed local resilience | Per-vertex in/out budgets | Directed Hamilton cycle | | Global resilience | Total number of deleted edges | Hamilton cycle | | \(H\)-resilience | Per-vertex loss of copies of \(H\) | Hamiltonian powers | | \(\mathbf{k}\)-resilience / degree-sequence resilience | Nonuniform or ordered deletion budgets | Hamilton cycle | | Mixed-fault endpoint resilience | Faulty vertices and edges | Hamiltonian or near-Hamiltonian path | ## 2. Random graphs and the random graph process The random graph process is the canonical setting in which Hamiltonian resilience became a sharp threshold phenomenon. A central theorem states that for every fixed \(\varepsilon>0\), asymptotically almost surely, for every \[ m\ge \frac{1+\varepsilon}{6}n\log n, \] the \(2\)-core of \(G_m\) is \((1/2-\varepsilon)\)-resilient with respect to Hamiltonicity [1710.00799]. At the hitting time \[ m_2=\min\{m:\delta(G_m)=2\}, \] the whole graph \(G_{m_2}\) is \((1/2-\varepsilon)\)-resilient with respect to Hamiltonicity [1710.00799]. A process-wide strengthening shows that, almost surely, every \(G_M\) with minimum degree at least \(2\) is \((1/2-\varepsilon)\)-resiliently Hamiltonian, but not \((1/2+\varepsilon)\)-resiliently Hamiltonian [1710.00505]. In this sense, Hamiltonicity in the random graph process is born resilient. The constant \(1/2\) is asymptotically optimal. A standard cut argument shows that one can delete about half of the incident edges at every vertex and leave a bipartite obstruction, so resilience above \(1/2\) is impossible in general [1710.00799]. The same paper shows that the resilient object below the minimum-degree-\(2\) hitting time is the \(2\)-core rather than the whole graph, because the whole graph may still contain vertices of degree \(0\) or \(1\). A complementary perspective comes from global resilience. Above the Hamiltonicity threshold, \[ np-\log n-\log\log n\to +\infty, \] the global resilience of Hamiltonicity in \(G(n,p)\) is exactly \[ r_g(G,\mathcal H)=\delta(G)-1 \] with high probability [2206.15235]. Thus the cheapest way to destroy Hamiltonicity globally is to lower a minimum-degree vertex to degree \(1\), even though locally the graph can survive deletion of nearly half the incident edges at every vertex. Near threshold, more degree-sensitive forms of resilience become relevant. One \(\mathbf{k}\)-resilience theorem shows that when \[ p\ge \frac{\ln n+\ln\ln n+\omega(1)}{n}, \] removing from each small-degree vertex all incident edges but two and from every other vertex at most a \((1/3-\varepsilon)\)-fraction of its incident edges still leaves a Hamiltonian graph [1101.3099]. In the denser sparse regime \(p\ge C\log n/n\), this implies \[ r_\ell(G,HAM)\ge (1-\varepsilon)\frac{np}{3}, \] while in the range up to \(1.02\log n/n\) it yields the exact local resilience value \[ r_\ell(G,HAM)=\delta(G)-1 \] with high probability [1101.3099]. ## 3. Beyond uniform local budgets: degree sequences, motifs, and Hamiltonian powers Uniform per-vertex deletion budgets do not exhaust the resilience landscape. A resilience version of Pósa’s theorem shows that for every \(\varepsilon>0\) there exists \(C>0\) such that, for \[ p\ge C\log n/n, \] asymptotically almost surely \(G_{n,p}\) is \(\varepsilon\)-Pósa-resilient with respect to Hamiltonicity [1810.12433]. In survivor-graph language, if the ordered degrees satisfy \[ d_i \ge (i+o(n))p \qquad \text{for all } i0\) such that if \[ p \geq C\log^3 n/\sqrt{n}, \] then with high probability every subgraph \(G\subseteq G_{n,p}\) in which each vertex remains on at least a \[ \left(\frac49+o(1)\right) \] fraction of its triangles contains the square of a Hamilton cycle, and the constant \(4/9\) is optimal [1809.07534]. This identifies the correct local object of robustness for \(C_n^2\): not edges, but triangles. A related almost-spanning result concerns squares of long cycles. For every \(\gamma,\nu>0\) and \[ p=n^{-1/2+\gamma/2}, \] asymptotically almost surely every spanning subgraph of \(G_{n,p}\) with minimum degree at least \[ (2/3+\nu)np \] contains the square of a cycle on at least \((1-\nu)n\) vertices [1606.02958]. The same work emphasizes that one cannot hope for a resilience version for the square of a spanning cycle in sparse random graphs, because deleting all edges in the neighborhood of a single vertex destroys that property [1606.02958]. This marks a structural difference between ordinary Hamiltonicity and higher powers. ## 4. Sparse structured hosts: regular, pseudorandom, geometric, and cycle-power settings Random regular graphs realize the Dirac threshold in a sparse regular host. For every \(\varepsilon>0\), there exists \(D\) such that, for \[ D0\), there exists \(C>0\) such that if \(G\) is a spanning subgraph of an \((n,d,\lambda)\)-graph with \[ \delta(G)\ge \left(\frac12+\gamma\right)d \qquad\text{and}\qquad \frac d\lambda \ge C, \] then \(G\) contains a Hamilton cycle [2507.22807]. The proof proceeds through a sparse upper-density condition, a cyclic partition into balanced bipartite expanders, and Hamilton-connectedness inside those pieces [2507.22807]. This removes the earlier polylogarithmic dependence of \(d/\lambda\) on \(n\). For powers of cycles, Hamiltonian resilience takes the form of a Dirac theorem inside a non-complete host. For every \(\varepsilon>0\) and all sufficiently large \(k\), every spanning subgraph \(G\subseteq C_n^k\) with \[ \delta(G)\ge (1+\varepsilon)k \] contains a Hamilton cycle [2606.07471]. Since \(\delta(C_n^k)=2k\) in the relevant regime, this means all \((1/2+\varepsilon)\)-subgraphs of \(C_n^k\) are Hamiltonian [2606.07471]. The theorem asymptotically settles a conjecture of Espuny Díaz, Lichev, and Wesolek, while the exact threshold \(k+1\) remains open [2606.07471]. Random geometric graphs exhibit a different behavior. For every fixed dimension \(d\), there is a positive Hamiltonicity resilience constant \[ \alpha(d)=\frac{1}{2d^{d/2}\theta_d}<\frac12 \] such that, for \[ C(\log n/n)^{1/d}\le r=o(1), \] asymptotically almost surely every \[ \Bigl(1-\frac{1}{2d^{d/2}\theta_d}+\varepsilon\Bigr)\text{-subgraph} \] of \(G_d(n,r)\) is Hamiltonian [2406.09921]. In dimension \(1\), this improves to the statement that every \((2/3+\varepsilon)\)-subgraph is Hamiltonian for \[ C\log n/n\le r=o(1) \] [2406.09921]. By contrast, the same paper proves \((1/2-\varepsilon)\)-resilience only for long cycles up to length \(2n/3\), and states that proving \((1/2-\varepsilon)\)-resilience for Hamiltonicity remains elusive with its techniques [2406.09921]. ## 5. Directed, hypergraph, and fault-tolerant network variants In random directed graphs, Hamiltonian resilience is measured against separate in- and out-degree deletions. For the binomial random digraph \({\mathcal D}(n,p)\), if \[ p \gg \frac{\log n}{\sqrt n}, \] then asymptotically almost surely \[ r_\ell({\mathcal D}(n,p),{\mathcal H})=(1/2+o(1))np, \] equivalently every spanning subdigraph with \[ \delta^+(D'),\delta^-(D') \ge (1/2+o(1))np \] contains a directed Hamilton cycle [1404.4734]. A subsequent improvement reduces the density requirement to \[ p=\omega\!\left(\frac{\log^8 n}{n}\right), \] which is optimal up to the polylogarithmic factor, while preserving the same asymptotic \(1/2\) resilience constant [1410.2198]. Together these results supply directed analogues of Ghouila-Houri-type robustness in sparse random settings. In random uniform hypergraphs, the relevant degree parameter is minimum codegree rather than minimum vertex degree. For every \(\gamma>0\) and \(k\ge 3\), if \[ p\ge n^{-1+\gamma}, \] then asymptotically almost surely every subgraph \(G\subseteq G^{(k)}(n,p)\) with \[ \delta_{k-1}(G)\ge \Big(\frac12+2\gamma\Big)pn \] contains a tight Hamilton cycle [2105.04513]. This is a codegree-resilience theorem for tight Hamiltonicity, asymptotically sharp in the degree threshold, and a hypergraph analogue of Dirac-type Hamiltonian resilience [2105.04513]. Fault-tolerant interconnection networks use a more endpoint-sensitive notion. In an \(n\)-dimensional twisted hypercube-like network \(G\), where \(n\ge 7\) and \[ |F|\le 2n-10 \] for a fault set \(F\subseteq V(G)\cup E(G)\), the surviving graph \(G-F\) has the following property: for any two surviving vertices \(s,t\) satisfying \[ N_{G-F}(s)-\{t\}\neq \emptyset \qquad\text{and}\qquad N_{G-F}(t)-\{s\}\neq \emptyset, \] there exists a Hamiltonian or near-Hamiltonian path between \(s\) and \(t\) in \(G-F\) [1111.5391]. Because twisted hypercube-like networks include crossed cubes, Möbius cubes, twisted cubes, and locally twisted cubes, this theorem is a unified endpoint-preserving resilience statement for several major hypercube variants [1111.5391]. The near-Hamiltonian fallback corresponds to concentrated faults creating a vertex of residual degree \(0\) or \(1\) in one recursive half of the network [1111.5391]. ## 6. Sharp constants, obstructions, and a distinct quantum-dynamical usage A consistent feature of the graph-theoretic literature is that the critical constant is often \(1/2\), but the reason depends on the model. In undirected random graphs and random regular graphs, a balanced-cut obstruction shows that deleting about half of the edges at every vertex can leave a bipartite graph with no Hamilton cycle [1710.00799], [1903.05052]. In random digraphs, deleting one crossing direction across a nearly balanced partition gives the same \(1/2\) barrier for in- and out-neighborhoods [1404.4734], [1410.2198]. In sparse pseudorandom graphs, the minimum-degree threshold near \(d/2\) is best possible, but the main obstruction is not mere expansion failure: the 2025 pseudorandom result isolates small-to-medium dense spots as the structural obstruction that must be excluded or controlled [2507.22807]. Other settings have different sharp constants. Triangle-resilience for the square of a Hamilton cycle has the optimal retained-triangle threshold \(4/9\) [1809.07534]. For almost-spanning square cycles in \(G_{n,p}\), the minimum-degree barrier is \(2/3\) [1606.02958]. For powers of cycles \(C_n^k\), the asymptotic threshold is \((1+o(1))k\), while the exact conjectured threshold is \(k+1\) [2606.07471]. In twisted hypercube-like networks, the mixed-fault threshold is linear in dimension, \[ |F|\le 2n-10, \] and the conclusion is prescribed-endpoint Hamiltonian or near-Hamiltonian connectivity rather than a single global Hamilton cycle [1111.5391]. The open problems follow the same pattern. Among them are the optimal dependence of \(C\) on \(\gamma\) in sparse pseudorandom graphs [2507.22807], extending directed resilience from \(\omega(\log^8 n/n)\) to the natural \(\omega(\log n/n)\) scale [1410.2198], proving \((1/2-o(1))\)-resilience for Hamiltonicity in random geometric graphs [2406.09921], establishing shifted Chvátal resilience for Hamiltonicity [1810.12433], and studying fault-tolerant pancyclicity, panconnectivity, and richer embeddings such as meshes and tori in twisted hypercube-like networks under the large fault model [1111.5391]. Outside graph theory, the phrase has a distinct quantum-dynamical usage. A 2026 paper studies resilience of unitary quantum dynamics under local Hamiltonian perturbations and proves entropy-dependent error bounds of the form \[ \|(U_0(t)-U(t))|\psi(0)\rangle\| \le t\|H_{\text{pert}}\|_F + \int_0^t d\tau\,\sqrt{\Delta_{H_{\text{pert}}^\dagger H_{\text{pert}}(\psi(\tau))}}, \] where the correction term decreases as subsystem entanglement entropy approaches its maximum [2606.07471]. That usage concerns robustness of dynamics generated by perturbed Hamiltonians rather than robustness of Hamiltonian cycles, but it preserves the same core idea: resilience is the persistence of a Hamiltonian object under structured perturbation [2606.07471]. Source: https://www.emergentmind.com/topics/hamiltonian-resilience