--- title: Random Regular Graphs url: https://www.emergentmind.com/topics/random-regular-graph type: topic --- # Random Regular Graphs Searching arXiv for recent and foundational papers on random regular graphs to ground the article. A random regular graph is a random graph in which every vertex has the same degree. In the standard uniform model \(G(n,d)\), one chooses uniformly from all simple \(d\)-regular graphs on the labeled vertex set \([n]=\{1,\dots,n\}\); equivalently, one may use the configuration model, in which each vertex receives \(d\) half-edges, a uniform perfect matching of the \(nd\) half-edges is sampled, and conditioning on simplicity yields the uniform law on simple \(d\)-regular graphs [1511.01175, 2503.10499]. Because typical instances are sparse, homogeneous, and locally tree-like, random regular graphs form a canonical setting for asymptotic enumeration, exact and approximate sampling, extremal subgraph questions, interacting particle systems, and disordered quantum models [1908.11388, 1712.00367]. ## 1. Ensemble, configuration model, and local weak limit For a simple graph \(G=(V,E)\) on \([n]\), \(d\)-regularity means \[ \deg_G(v)=d \qquad \text{for all } v\in V. \] The state space \(\mathcal G_{n,d}\) consists of all simple \(d\)-regular graphs on \([n]\), and the uniform random \(d\)-regular graph is the uniform distribution on \(\mathcal G_{n,d}\) [1511.01175]. A broader notation \(\mathcal G(n,\mathbf d)\) is used for a prescribed degree sequence \(\mathbf d=(d_1,\dots,d_n)\); the regular case is \(\mathbf d=(d,\dots,d)\) [1906.02886]. The configuration model realizes the same object before conditioning on simplicity. In the formulation used for the random \(d\)-regular configuration model, one partitions \([dN]\) into \(N\) blocks of size \(d\), chooses a uniform perfect matching of the \(dN\) half-edges, and interprets each matched pair as an edge between the corresponding vertices; the result is a random \(d\)-regular multigraph on \([N]\) [2503.10499]. This multigraph is simple with asymptotically positive probability, so high-probability statements proved in the configuration model transfer to the uniform simple model with positive limiting probability [2503.10499]. A central structural fact is local convergence to the infinite \(d\)-regular tree \(\mathbb T_d\). For a uniformly chosen vertex \(i\in[N]\), the radius-\(r\) ball \(B_G(i,r)\) converges in distribution to the radius-\(r\) ball around the root in \(\mathbb T_d\); the same remains valid for \(r=\varepsilon\log N\) with small fixed \(\varepsilon>0\) [2503.10499]. This local weak limit is the basis for transferring tree-based recursion, cavity, and branching arguments to large finite random regular graphs. ## 2. Local geometry, expansion, and effective tree structure Random regular graphs are locally tree-like in a stronger asymptotic sense than local weak convergence alone. For asymptotically almost every \(G\sim\mathcal G(n,d)\), all but \(o(n)\) vertices have neighborhoods of radius \(\lfloor \log_{d-1}\log n\rfloor\) that are trees, and the diameter is \((1+o(1))\log_{d-1} n\) [1503.04895]. The same paper uses edge-expansion estimates of the form \[ \Psi_E(G,\varepsilon n)\ge d-2-\delta \] for asymptotically almost every \(d\)-regular graph, together with a positive isoperimetric constant, to control spreading processes and metastability [1503.04895]. For random regular graphs of connectivity \(K+1=3\), the number of vertices at graph distance \(x\) from a given vertex grows as \[ N(x)\sim K^x, \] and the typical distance between two random vertices scales as \[ D\sim \frac{\ln L}{\ln K}. \] At distances \(x\) much smaller than the diameter, such graphs look like regular trees with branching number \(K\), while loops are typically long [1908.11388]. This makes random regular graphs a precise, controllable setting for “infinite-dimensional” geometry without the boundary effects of the Bethe lattice: they are finite and homogeneous, and all vertices are equivalent [1908.11388]. In dense degree regimes, random regular graphs also fall into robust-expander classes. In particular, graphs on \(n\) vertices with minimum degree at least \((\tfrac12+o_n(1))n\) are robust expanders, so a random \(d\)-regular graph with \(d>(\tfrac12+o(1))n\) satisfies this condition with high probability [2301.10131]. This robust-expansion viewpoint underlies asymptotic results on perfect matchings and Poisson overlap statistics. ## 3. Sampling, counting, and coupling with binomial random graphs Algorithmic generation of uniform or nearly uniform random regular graphs has been a longstanding theme. Bayati, Kim, and Saberi gave a nearly-linear-time sequential algorithm for graphs with a prescribed degree sequence when \[ d_{\max}=O(m^{1/4-\tau}), \] producing almost uniform samples in time \(O(m\,d_{\max})\); for regular graphs, they showed that if \[ d=O(n^{1/2-\tau}), \] their algorithm generates an asymptotically uniform \(d\)-regular graph, improving the previous \(d=O(n^{1/3-\tau})\) range due to Kim and Vu [0702124]. A different line, based on switchings in the configuration model, yielded exact and approximate uniform samplers with explicit complexity guarantees. The sampler REG generates \(d\)-regular graphs uniformly at random in expected time \[ O(nd^3) \] for \(d=o(n^{1/2})\), while the approximate sampler REG\(^*\) runs in expected time \[ O(dn) \] and outputs a \(d\)-regular graph whose distribution differs from uniform by \(o(1)\) in total variation distance, again for \(d=o(n^{1/2})\) [1511.01175]. The exact algorithm preserves uniformity by equalizing expected visit counts across strata of pairings, and the approximate version drops the rejection machinery while retaining asymptotic correctness [1511.01175]. Coupling random regular graphs to binomial random graphs provides a complementary transfer principle. Kim and Vu conjectured that if \(d\gg \log n\), then \(\mathcal G(n,d)\) should be asymptotically sandwichable between two Erdős–Rényi graphs with edge probabilities \((1+o(1))d/n\). Gao, Isaev, and McKay proved the conjecture with perfect containment on both sides for all \[ d\gg \frac{n}{\sqrt{\log n}}, \] and for smaller \(d=O(n/\sqrt{\log n})\) obtained a weaker sandwich in which the upper parameter is approximately \((d/n)\log n\) [1906.02886]. In the near-regular setting this becomes a general tool for transferring Hamiltonicity, universality, chromatic number, small subgraph counts, diameter, independence number, and edge-percolation thresholds from \(\mathcal G(n,p)\) to \(\mathcal G(n,\mathbf d)\) [1906.02886]. ## 4. Factors, perfect matchings, and sprinkling Uniform perfect matchings in dense regular robust expanders exhibit strong edge-uniformity. If \(G\) is a \(d\)-regular robust \((\nu,\tau)\)-expander on \(n\) vertices, \(M\) is a uniformly chosen perfect matching of \(G\), and \(e\in E(G)\), then \[ \mathbb P[e\in M]=(1+o_n(1))\,d^{-1}. \] Moreover, for any fixed matching \(N\subseteq E(G)\), the random variable \(|M\cap N|\) is asymptotically Poisson with parameter \(d^{-1}|N|\), and for a spanning \(r\)-regular subgraph \(N\subseteq G\) with \(r\le n^{1/50}\), \(|M\cap E(N)|\) is asymptotically Poisson with parameter \(rn/(2d)\) [2301.10131]. Since random \(d\)-regular graphs with \(d>(\tfrac12+o(1))n\) are robust expanders with high probability, these asymptotics describe the typical dense random-regular case as well [2301.10131]. A more structural decomposition question is whether the edge-disjoint union of two random regular graphs behaves like a random regular graph of the combined degree. The conjectural statement is that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of degree \(d_1+d_2\), provided the combined degree grows with \(n\). This has been verified in several regimes: when \(d_1=1\), \(d_2\to\infty\), and \(d_2=o(n^{1/3})\); when \[ \min\{d_1,d_2,n-d_1-d_2\}=\omega(n/\log n); \] and in a dense–sparse regime specified by \[ \frac{d_2^2}{\min\{d_2,n-d_2\}} \le \frac{2d_1}{3\log n}. \] The same work also proves an asymptotic formula for the expected number of spanning regular subgraphs in a random regular graph [2309.00190]. The reduction underlying these sprinkling results identifies concentration of \[ |\mathcal R_{d_1}(G_{d_1+d_2})| \] —the number of spanning \(d_1\)-regular subgraphs of a random \((d_1+d_2)\)-regular graph—as equivalent to the desired coupling between the union model and \(G(n,d_1+d_2)\) [2309.00190]. This is a distinctly regular-graph analogue of the classical sprinkling method for \(G(n,p)\). A plausible implication is that many multi-round exposure arguments in the Erdős–Rényi setting should have regular-graph counterparts whenever factor-count concentration is available. ## 5. Propagation processes, cover times, and finite-graph phase transitions Bootstrap percolation on random regular graphs leads to an extremal optimization problem: the size of a minimal contagious set. For a \((k+1)\)-regular random graph with threshold \(l\), the cavity method yields several quantitative conjectures, notably \[ \theta_{\min}(4,3)=\frac16,\qquad \theta_{\min}(5,4)=\frac14, \] meaning that 5-regular random graphs with threshold 3, and 6-regular random graphs with threshold 4, have contagious sets of asymptotic density \(1/6\) and \(1/4\), respectively [1407.7361]. Equivalently, these are the minimal fractions of vertices that have to be removed from a 5-regular random graph to destroy its 3-core, and from a 6-regular random graph to destroy its 4-core [1407.7361]. For the contact process on a uniform random \(d\)-regular graph with \(d\ge3\), initialized from full occupancy, the extinction time grows exponentially with the vertex number when \(\lambda>\lambda_1(\mathbb T^d)\), and the normalized extinction time converges in distribution to \(\mathrm{Exp}(1)\) for asymptotically almost every graph [1503.04895]. A more recent result identifies a finite-graph analogue of the second tree critical value \(\lambda_2(d)\). On the random \(d\)-regular configuration model on \([N]\), started from \(\xi_0=\{1\}\), let \(I_k\) be the time of the \(k\)-th reinfection of vertex \(1\). Conditional on survival for \(\omega(1)\) time, for every \(\epsilon\in(0,1)\), \[ I_{\lfloor\log^\epsilon N\rfloor}=\Theta_{\mathbb P}(\log^\epsilon N)\quad\text{if }\lambda>\lambda_2(d), \] whereas \[ I_{\lfloor\log^\epsilon N\rfloor}=\frac{1}{c_\lambda}\log N+\Theta_{\mathbb P}(\log^\epsilon N)\quad\text{if }\lambda_1(d)<\lambda<\lambda_2(d), \] with \(c_\lambda\) the Malthusian growth parameter on \(\mathbb T_d\) [2503.10499]. This identifies the second transition as one between linear reinfections and reinfections following a long healthy period. Random-walk exploration also acquires explicit asymptotics on random regular graphs. For the biased random walk that always prefers previously unvisited edges, on a random \(r\)-regular graph \(G_r\) with fixed odd \(r\ge3\), \[ C_V(G_r)\sim \frac{1}{r-2}n\log n,\qquad C_E(G_r)\sim \frac{r}{2(r-2)}n\log n \] with high probability, for vertex and edge cover times respectively [1805.05780]. In parallel, the random-cluster Glauber dynamics on random \(\Delta\)-regular graphs mixes in \(\Theta(n\log n)\) for every \(q\ge1\), \(\Delta\ge3\), and \(p