Quasi-Random Graphs

• Quasi-random graphs are deterministic graph sequences that mimic the edge distribution, subgraph counts (like C4 counts), and spectral properties of G(n,p) random graphs.
• They encompass techniques from counting methods to graphon analysis, forcing families, and cut properties to establish a full random-like structure from local constraints.
• Recent studies extend the theory through explicit algebraic constructions, hypergraph analogues, and dynamic characterizations, refining our quantitative and structural understanding.

Quasi-random graphs are deterministic graphs, or graph sequences, that reproduce the random-looking edge distribution, subgraph counts, and spectral behaviour of the binomial random graph $G(n,p)$. The subject originates in the observation that several apparently different properties of $G(n,p)$ are asymptotically equivalent in dense graphs, and it now encompasses forcing families, graphon formulations, cut properties, oriented and hypergraph analogues, explicit algebraic constructions, and even dynamical characterizations via interacting Markov processes. A central lesson is that some sparse-looking tests, such as edge and $C_4$ counts, force full random-like structure, while other apparently strong data, including all clique densities, may fail to do so (Shapira et al., 2021).

1. Classical dense quasi-randomness

Fix $0$H=(V(H),E(H))$, standard probabilistic arguments show that with high probability $G(n,p)$ contains

$(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$

labelled copies of $H$. Chung, Graham, and Wilson showed that this subgraph-count behaviour is equivalent to several other properties. In the formulation emphasized here, an $n$-vertex graph $G$ is $G(n,p)$0-quasirandom if it satisfies any, hence all, of the following: it has $G(n,p)$1 labelled edges and $G(n,p)$2 labelled copies of $G(n,p)$3; for every fixed graph $G(n,p)$4, it has $G(n,p)$5 labelled copies of $G(n,p)$6; and for every $G(n,p)$7 and every $G(n,p)$8 with $G(n,p)$9, the number of edges inside $C_4$0 is $C_4$1 (Shapira et al., 2021).

This equivalence is the foundational phenomenon of quasi-random graph theory. It says that a weak local constraint, such as the correct edge density together with the correct number of $C_4$2-cycles, already forces the full small-subgraph profile and uniform edge distribution. In a closely related formulation, Chung–Graham–Wilson properties $C_4$3, $C_4$4, and $C_4$5 identify the same class of $C_4$6-quasi-random graph sequences through edge counts in all subsets, edge counts in subsets of fixed density, and the joint control of total edges and $C_4$7-counts (Huang et al., 2010).

2. Cut properties, restricted subgraph counts, and graphons

A major extension of the classical theory studies not just global counts of a fixed graph $C_4$8, but counts subject to geometric restrictions on where the vertices of $C_4$9 may lie. For a partition $0$H=(V(H),E(H))$0, where the perfectly balanced cut property is not quasi-random (Huang et al., 2010).

The graphon formalism translates such questions into analytic identities. For a graphon $H=(V(H),E(H))$1 and a labelled graph $H=(V(H),E(H))$2 on $H=(V(H),E(H))$3, one sets

$H=(V(H),E(H))$4

Restricted subgraph-count properties become statements about integrals of $H=(V(H),E(H))$5 over product sets of prescribed measure, and $H=(V(H),E(H))$6-quasi-randomness becomes the assertion that the only graphon satisfying the relevant identities is the constant graphon $H=(V(H),E(H))$7. This graph-limit translation underlies both the Simonovits–Sós style subset-count characterizations and later refinements involving disjoint sets, repeated sets, and balanced versus unbalanced parameter regimes (Janson et al., 2014).

Recent work sharpens this analysis by decomposing the symmetric subspace of $H=(V(H),E(H))$8 into irreducible subspaces under the action of measure-preserving bijections of $H=(V(H),E(H))$9. This makes it possible to isolate “bad” parameter values $G(n,p)$0 for which nontrivial symmetric components survive the integral constraints, and to separate the representation-theoretic part of the problem from the algebraic problem of deciding whether a given graph $G(n,p)$1 can realize those components through $G(n,p)$2. This suggests that many of the exceptional cases are not accidental, but are controlled by a precise invariant-subspace structure (Janson, 18 Jun 2026).

3. Forcing families and the failure of clique counts

A family $G(n,p)$3 of finite graphs is called forcing if, for every $G(n,p)$4, every graph sequence with asymptotically correct labelled counts of every $G(n,p)$5 is $G(n,p)$6-quasi-random. The theorem of Chung–Graham–Wilson says that $G(n,p)$7 is forcing. By contrast, $G(n,p)$8 is not forcing for any non-bipartite $G(n,p)$9, and infinite families such as all cycles, all stars, or all trees are also not forcing. The Skokan–Thoma conjecture asks whether $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$0 is forcing whenever $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$1 is a connected bipartite graph that is not a tree (Shapira et al., 2021).

A particularly striking negative result is that the family of all cliques is not forcing. For finite clique families $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$2, one can construct complete $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$3-partite graphs whose $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$4-counts match those of $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$5 for every $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$6, yet which have an independent set of size at least $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$7, and hence are far from quasirandom. More strongly, for every $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$8, there exists a graph sequence $(1+o(1))\,p^{|E(H)|}n^{|V(H)|}$9 such that for every $H$0,

$H$1

but $H$2 is not $H$3-quasirandom (Shapira et al., 2021).

The analytic mechanism behind this counterexample passes through the pantograph equation

$H$4

whose solution is the deformed exponential

$H$5

The zeros of $H$6 yield positive coefficients $H$7 such that the complete infinite-partite graphon with parts of measures $H$8 satisfies

$H$9

while still containing an interval of length at least $n$0 on which $n$1. The misconception that “matching every clique density should force randomness” is therefore false: clique densities do not detect large multipartite or independent-block structure (Shapira et al., 2021).

4. Directed and hypergraph analogues

The quasi-random paradigm extends beyond simple undirected graphs, but the equivalence pattern becomes subtler. For oriented graphs with arbitrary underlying graph $n$2, Griffiths proved that nine natural random-orientation conditions are equivalent, including balanced edge counts across cuts, correct counts of all small oriented graphs, and spectral conditions on the skew-adjacency matrix. Among the four orientation types of a $n$3-cycle, exactly two—Type II and Type IV—yield quasi-randomness tests: if their counts are close to the random-orientation expectation, then the same is true for every small oriented graph (Griffiths, 2011).

For $n$4-uniform hypergraphs, quasi-randomness is typically phrased through $n$5-denseness. In the $n$6-partite setting this means

$n$7

for all $n$8. Unlike the graph case, different hypergraph quasi-randomness notions are not generally equivalent. In the factor problem, however, a powerful reduction is available: for every $n$9,

$G$0

so perfect $G$1-tilings in quasi-random $G$2-graphs with minimum degree $G$3 are controlled by the apparently weaker problem of covering every vertex by some copy of $G$4. This yields a characterization for $G$5-partite $G$6-graphs and, separately, a complete characterization of $G$7 using the Reiher–Rödl–Schacht description of $G$8-graphs with vanishing Turán density in quasi-random $G$9-graphs (Ding et al., 2021).

A complementary partite theory shows that if $G(n,p)$00 and $G(n,p)$01 is a fixed balanced $G(n,p)$02-partite $G(n,p)$03-graph, then any sufficiently large $G(n,p)$04-dense $G(n,p)$05-partite $G(n,p)$06-graph with partite minimum codegree $G(n,p)$07 contains an $G(n,p)$08-factor, and that $G(n,p)$09 is best possible. Moreover, $G(n,p)$10 is the asymptotic partite minimum codegree threshold for all fixed balanced $G(n,p)$11-partite $G(n,p)$12-graph factors even without quasi-randomness, while lower partite minimum $G(n,p)$13-degree conditions with $G(n,p)$14 do not suffice (Sun, 2023).

The hypergraph setting also exhibits a sharp contrast with graphs in Hamiltonian counting. For graphs, it remains open whether a quasi-random graph of density $G(n,p)$15 can exceed the random expectation $G(n,p)$16 by a polynomial factor. For $G(n,p)$17-uniform hypergraphs with $G(n,p)$18, by contrast, there exist quasi-random $G(n,p)$19-graphs with density $G(n,p)$20 whose number of Hamiltonian cycles is larger than $G(n,p)$21 by an exponential factor. This shows that weak hypergraph quasi-randomness does not control Hamilton cycle counts in the way graph quasi-randomness often controls spanning structure (Yuster, 2022).

5. Quantitative quasi-randomness and explicit constructions

One quantitative viewpoint encodes graph and hypergraph discrepancy through multilinear Littlewood polynomials. In degree $G(n,p)$22, the polynomial associated with a graph $G(n,p)$23 is

$G(n,p)$24

The Chung–Graham–Wilson cut-discrepancy condition becomes the requirement that $G(n,p)$25. More generally, for a $G(n,p)$26-bounded multilinear Littlewood family $G(n,p)$27, the extremal sup norm is controlled by

$G(n,p)$28

between a lower and upper bound of order $G(n,p)$29 for fixed $G(n,p)$30, where $G(n,p)$31 is the degree of vertex $G(n,p)$32 in the support hypergraph. In the graph case $G(n,p)$33, this gives $G(n,p)$34, and the Paley graph attains the correct order (Kalai et al., 2018).

A second quantitative framework is the bi-jumbled condition $G(n,p)$35, defined by

$G(n,p)$36

for all $G(n,p)$37. No family can satisfy $G(n,p)$38 with $G(n,p)$39. Paley graphs and Paley sum graphs satisfy $G(n,p)$40, and recent polynomial constructions produce broader “Paley-like” families that are neither Cayley graphs nor Cayley sum graphs. For admissible polynomials $G(n,p)$41, the graphs $G(n,p)$42 form families $G(n,p)$43 with $G(n,p)$44; a substantial subfamily $G(n,p)$45, obtained from admissible $G(n,p)$46 that are linear in each variable and compositions $G(n,p)$47, has $G(n,p)$48. These families include Paley graphs, Paley sum graphs, and Diophantine graphs $G(n,p)$49 defined by the rule that $G(n,p)$50 and $G(n,p)$51 are adjacent iff $G(n,p)$52 is a square (Kim et al., 2024).

The $G(n,p)$53 framework also yields logarithmic lower bounds on clique and independence numbers. In particular, if a family has $G(n,p)$54 and satisfies the sufficient condition used there, then the clique number is at least

$G(n,p)$55

This applies to many classical quasi-random graphs, including Paley graphs and Paley sum graphs, as well as to the new polynomial constructions (Kim et al., 2024).

6. Dynamics and recent structural directions

Quasi-randomness also admits a dynamical characterization. For Markov population processes on graph sequences, with local transition rates affine in the neighborhood state vector, the homogeneous mean-field approximation is the ODE

$G(n,p)$56

It is accurate on every finite time horizon if and only if the underlying graph sequence is quasi-random in the discrepancy sense

$G(n,p)$57

Moreover, the approximation error is of order $G(n,p)$58 plus the largest discrepancy of the graph, and for Erdős–Rényi and random regular graphs it is of order the inverse square root of the average degree. Diverging average degree is therefore a necessary condition for HMFA accuracy in this framework (Keliger, 2021).

A recent structural development revisits regularity-type decomposition itself. For approximately degree-regular bipartite graphs, one can partition the vertex set into a bounded number of large vertex-disjoint quasi-random pairs, called bundles, together with a small exceptional set, without invoking Szemerédi’s Regularity Lemma. An $G(n,p)$59-bundle is an $G(n,p)$60-regular pair in which every vertex has degree $G(n,p)$61. The resulting decomposition remains effective even as the density tends to zero and for vertex sizes far smaller than those usually required by the Regularity Method. This suggests a narrower but quantitatively milder route to quasi-random structure, especially for bipartite packing problems (Csaba, 24 May 2026).

Quasi-random graph theory is therefore best understood not as a single theorem, but as a network of equivalences, forcing phenomena, exceptions, and transfer principles. In graphs, $G(n,p)$62-counts, subset counts, cut properties, spectral gaps, and graphon constancy are deeply interchangeable; in oriented graphs and hypergraphs, some of that equivalence survives but new obstructions appear; and in recent work, algebraic constructions, dynamical limits, and alternative decomposition methods continue to enlarge both the scope and the boundaries of the subject.