Weak Saturation Number in Graph and Hypergraph Theory

• Weak saturation number is defined as the minimum number of edges in a graph or hypergraph from which adding each missing edge creates a new copy of a fixed subgraph.
• Asymptotic analyses yield rational limit laws and exact formulas for specific cases such as cliques, hypercubes, and balanced multipartite hosts.
• Researchers use algebraic, polymatroid, and probabilistic methods to establish lower bounds, study random-host stability, and address computational complexity.

Searching arXiv for recent and foundational papers on weak saturation number and related variants. I checked the recent arXiv corpus for “weak saturation” and related phrases; the core sources relevant here include hypergraph asymptotics (Terekhov, 4 Apr 2025), rational weak-saturation limits (Ascoli et al., 26 Jan 2025), computational complexity (Tancer et al., 21 Jan 2025), limits of linear-algebraic lower bounds (Terekhov et al., 2024), random-host stability (Kalinichenko et al., 2023, Kalinichenko et al., 2021, Bidgoli et al., 2020), tree-specific behavior (Chen et al., 21 Feb 2025), hypercube and grid formulas (Morrison et al., 2014), multipartite and tensor-product hypergraphs (Bulavka et al., 2021, Terekhov, 8 Apr 2026), and asymptotically optimal hypergraph lower bounds (Terekhov, 8 Apr 2026). The weak saturation number is an extremal parameter that measures how sparsely one can begin and still force a host graph or hypergraph to grow to completion by repeatedly adding edges that each create a new copy of a fixed pattern. In the classical complete-host setting, for an $r$-uniform hypergraph $H$, $\mathrm{wsat}(n,H)$ is the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices from which the complete $r$-graph can be obtained by adding the missing edges one at a time, each addition creating a new copy of $H$; in a general host $G$, one similarly studies $\mathrm{wsat}(G,H)$ for spanning subgraphs of $G$ (Terekhov, 4 Apr 2025, Kalinichenko et al., 2021). Originating in work of Bollobás and closely connected to bootstrap percolation, the notion now spans exact formulas, asymptotic limit laws, random-host stability, algebraic and polymatroidal lower bounds, hypergraph extensions, and colored variants (Morrison et al., 2014, Terekhov et al., 2024).

1. Definition and formal framework

For graphs $H$0 and $H$1, a spanning subgraph $H$2 is weakly $H$3-saturated in $H$4 if the missing edges of $H$5 in $H$6 can be added one by one in some order so that every added edge creates a new copy of $H$7 that contains that edge (Korándi et al., 2015, Tancer et al., 21 Jan 2025). The corresponding minimum edge count is

$H$8

In the complete-host case one writes $H$9, and for $\mathrm{wsat}(n,H)$0-uniform hypergraphs the analogous definition uses $\mathrm{wsat}(n,H)$1 and $\mathrm{wsat}(n,H)$2-edges throughout (Terekhov, 4 Apr 2025).

The process formulation is central. In the hypergraph setting, if $\mathrm{wsat}(n,H)$3 is an $\mathrm{wsat}(n,H)$4-uniform hypergraph on $\mathrm{wsat}(n,H)$5 and

$\mathrm{wsat}(n,H)$6

then weak $\mathrm{wsat}(n,H)$7-saturation means that there is a chain

$\mathrm{wsat}(n,H)$8

with $\mathrm{wsat}(n,H)$9 such that each $r$0 lies in a new copy of $r$1 in $r$2 (Terekhov, 4 Apr 2025). This distinguishes weak saturation from classical saturation, where every missing edge must create a copy immediately, rather than only after a suitable growth process (Korándi et al., 2015).

The general-host formulation and the complete-host formulation are complementary. The former emphasizes spanning subgraphs of an ambient graph or hypergraph $r$3, while the latter emphasizes the asymptotics of $r$4 for fixed $r$5. The literature also treats directed multipartite hosts, random hosts, grids, hypercubes, and colored hosts, but the same bootstrap principle remains the common core (Bulavka et al., 2021, Morrison et al., 2014).

2. Asymptotic theory in complete hosts

For graphs, Alon showed that for every fixed graph $r$6 there exists a constant $r$7 such that

$r$8

so the normalized quantity $r$9 always has a limit (Ascoli et al., 26 Jan 2025). Recent work goes further and characterizes all possible rational values of $n$0: they are exactly

$n$1

which in particular implies that $n$2 can equal any rational number at least $n$3 (Ascoli et al., 26 Jan 2025).

For $n$4-uniform hypergraphs, the asymptotic scale is governed by the sparseness parameter

$n$5

the minimum size of a vertex set contained in exactly one edge of $n$6 (Terekhov, 4 Apr 2025). Tuza proved the general upper bound

$n$7

and conjectured that this exponent is always exact. That conjecture is now proved: for every $n$8-uniform hypergraph $n$9 with at least two edges, there exists $r$0 such that

$r$1

(Terekhov, 4 Apr 2025).

A different asymptotic viewpoint uses minimum positive codegree. For an $r$2-uniform hypergraph $r$3, the parameter

$r$4

generalizes minimum degree. If $r$5, then

$r$6

and this lower bound is asymptotically sharp in general (Terekhov, 8 Apr 2026). In the graph case $r$7, this recovers the lower bound

$r$8

(Terekhov, 8 Apr 2026).

3. Exact formulas and representative families

Several natural host-pattern pairs admit exact weak saturation numbers.

Family	Host and pattern	Result
Cliques	$r$9 vs. $H$0	$H$1 (Korándi et al., 2015)
Hypercubes	$H$2 vs. $H$3	$H$4 (Morrison et al., 2014)
Grid cycles	$H$5 vs. $H$6	$H$7 for $H$8 (Morrison et al., 2014)
Balanced multipartite $H$9-graphs	$G$0 vs. $G$1	exact multipartite formula (Bulavka et al., 2021)

For cliques in complete hosts, weak and strong saturation coincide: $G$2 a phenomenon that is highly specific to this setting (Korándi et al., 2015). For hypercubes, the exact formula for $G$3 answers a question of Johnson and Pinto and shows that, for fixed $G$4, the weak saturation number is $G$5 (Morrison et al., 2014).

Complete multipartite hypergraphs admit an exact directed formula. If $G$6, $G$7, and $G$8 is the complete $G$9-partite $\mathrm{wsat}(G,H)$0-graph, then the directed weak saturation number is

$\mathrm{wsat}(G,H)$1

which generalizes Alon’s $\mathrm{wsat}(G,H)$2 theorem; in the balanced case $\mathrm{wsat}(G,H)$3, this is also the undirected weak saturation number (Bulavka et al., 2021). The same paper proves that for fixed $\mathrm{wsat}(G,H)$4,

$\mathrm{wsat}(G,H)$5

so the leading term in the clique host depends only on the smallest part size (Bulavka et al., 2021).

Trees exhibit a different behavior because the host size $\mathrm{wsat}(G,H)$6 eventually ceases to matter. If $\mathrm{wsat}(G,H)$7 is a tree, then $\mathrm{wsat}(G,H)$8 is non-increasing in $\mathrm{wsat}(G,H)$9 and stabilizes for large $G$0, yielding a limiting weak saturation number $G$1 (Chen et al., 21 Feb 2025). For nondegenerate caterpillars $G$2 on $G$3 vertices, with $G$4, one has

$G$5

and otherwise

$G$6

moreover, for every $G$7 there are caterpillars on $G$8 vertices whose weak saturation numbers are $G$9 (Chen et al., 21 Feb 2025).

4. Methods of proof and structural principles

A large part of the theory is driven by algebraic lower bounds. Kalai’s linear-algebraic method assigns vectors to edges of the host so that every copy of the pattern gives a linear dependence; if every copy of $H$00 yields such a dependence, then the dimension of the span is a lower bound on $H$01 (Terekhov et al., 2024). This viewpoint is formalized via the weak saturation rank $H$02, the maximum rank of a matroid on $H$03 in which every copy of $H$04 is a circuit (Terekhov et al., 2024).

That method is powerful but not universal. For every graph $H$05, $H$06 is eventually of the form $H$07 with integer slope $H$08, and there are infinitely many graphs $H$09 for which this cannot match $H$10 asymptotically because the weak saturation limit is non-integer (Terekhov et al., 2024). The same paper proposes a multigraph modification of Kalai’s method that restores tight lower bounds for some such patterns, including dumbbell graphs (Terekhov et al., 2024).

For hypergraphs, polymatroids extend the algebraic method beyond integer coefficients. A 1-polymatroid is a real-valued monotone submodular rank function bounded by set size, and a weakly $H$11-saturated 1-polymatroid gives the same kind of lower bound as a matroid. The paper "Asymptotically optimal lower bounds on weak saturation numbers for hypergraphs" introduces a polymatroid lower-bound method based on count polymatroids and shows that, unlike the original linear-algebraic method, it can yield non-integer asymptotic coefficients (Terekhov, 8 Apr 2026).

The short proof of Tuza’s conjecture for hypergraphs shows a different proof paradigm. There the key structural lemma states that if an $H$12-uniform hypergraph $H$13 has vertex partition $H$14, $H$15, and contains all edges $H$16 with

$H$17

then $H$18 is weakly $H$19-saturated (Terekhov, 4 Apr 2025). Combined with Rödl’s covering theorem, this gives a compact transfer argument from small asymptotically optimal weakly saturated hypergraphs to large ones, avoiding the more elaborate template machinery used in the earlier proof by Shapira and Tyomkyn (Terekhov, 4 Apr 2025).

5. Random hosts and stability phenomena

A major theme is whether weak saturation numbers are stable under random thinning of the complete host. For every fixed graph $H$20 and every constant $H$21,

$H$22

so dense Erdős–Rényi hosts are asymptotically indistinguishable from complete hosts for weak saturation (Kalinichenko et al., 2023). For cliques this is sharper: if $H$23 and $H$24 is constant, then

$H$25

exactly the same value as in $H$26 (Korándi et al., 2015).

The random-host theory also reveals a sharp contrast with strong saturation. In $H$27 with constant $H$28, strong clique saturation is of order $H$29, whereas weak clique saturation remains linear and equal to the complete-host value (Korándi et al., 2015). This suggests that weak saturation is much more robust under host sparsification than classical saturation.

The mechanisms behind stability are structural. One approach uses $H$30-good graphs, where many small sets admit many clique extensions and large sets interact through forced cliques; random graphs $H$31 are $H$32-good with high probability for fixed $H$33 and constant $H$34, which yields exact weak clique saturation formulas (Korándi et al., 2015). A more general transference result shows that if a minimal weakly $H$35-saturated graph has a bounded “local percolating core” $H$36 such that every other vertex has at least $H$37 neighbors in $H$38, then

$H$39

with high probability for constant $H$40; this applies, for example, to cliques and complete bipartite graphs (Kalinichenko et al., 2021).

Below constant density, thresholds appear. For weak $H$41-saturation stability, the property

$H$42

has a threshold function $H$43, with lower bound of order

$H$44

and upper bound of order

$H$45

where $H$46 for $H$47 and $H$48 for $H$49 (Bidgoli et al., 2020). For stars $H$50, stability occurs at a different scale: $H$51 which is the threshold up to constants for

$H$52

when $H$53 (Kalinichenko et al., 2021).

6. Variants: colored, rainbow, and tensor-product settings

Several related parameters modify the weak saturation condition by incorporating colors or families of target hypergraphs. In the rainbow setting, a graph is weakly $H$54-rainbow saturated if its non-edges can be ordered so that, for any list of pairwise distinct colors, the edges can be added one by one and each added edge creates a new rainbow copy of $H$55 (Li et al., 2024). The corresponding weak rainbow saturation number $H$56 satisfies

$H$57

for every non-empty graph $H$58, and this limit is zero if and only if $H$59 contains a pendant edge (Li et al., 2024).

Rainbow and weak rainbow saturation diverge markedly from the uncolored theory. For complete graphs, the uncolored parameters satisfy

$H$60

but the rainbow theory separates: for $H$61, rainbow saturation and weak rainbow saturation are not asymptotically equal, and in fact the rainbow saturation number exceeds rainbow semisaturation, hence weak rainbow saturation, by a linear term (Chakraborti et al., 2022).

Colored hypergraph weak saturation also admits exact formulas in highly structured hosts. The paper "Weak saturation of tensor product of cliques" determines weak saturation numbers when both host and pattern are tensor products of cliques, generalizing the Moshkovitz–Shapira result for $H$62 versus $H$63. It also determines colored weak saturation numbers $H$64 for unions of tensor products of cliques and for arbitrary families of such target hypergraphs (Terekhov, 8 Apr 2026).

These variants indicate that the bootstrap-percolation principle extends well beyond the basic uncolored complete-host setting. A plausible implication is that weak saturation is best viewed as a family of related closure parameters rather than a single isolated invariant, with the uncolored $H$65 as the foundational case.

7. Complexity, limitations, and open directions

For arbitrary hosts, exact computation is difficult. Determining whether an $H$66-vertex graph $H$67 satisfies

$H$68

is NP-hard (Tancer et al., 21 Jan 2025). The proof builds a polynomial-time reduction from $H$69-SAT through shellability and collapsibility of pure $H$70-dimensional simplicial complexes, showing that even the extremal question of whether there is a weakly triangle-saturated spanning tree is algorithmically intractable (Tancer et al., 21 Jan 2025).

Several structural questions remain open. In the hypergraph asymptotic theorem

$H$71

the constant $H$72 is not determined in general, and exact values or finer error terms remain open even for specific families (Terekhov, 4 Apr 2025). Random-host stability is also incomplete: the exact threshold for

$H$73

is unknown, and a general conjecture asserts that for every fixed graph $H$74 and constant $H$75,

$H$76

(Bidgoli et al., 2020, Kalinichenko et al., 2021).

On the asymptotic side, the spectrum of weak saturation limits is only partially understood beyond the rational classification. One conjecture states that for every graph $H$77, the weak saturation limit $H$78 is rational (Ascoli et al., 26 Jan 2025). On the lower-bound side, the polymatroid framework raises the question whether the best polymatroid bound $H$79 always matches $H$80 asymptotically, or even exactly (Terekhov, 8 Apr 2026).

Taken together, these directions show that the weak saturation number sits at the intersection of extremal graph theory, hypergraph asymptotics, probabilistic combinatorics, algebraic methods, and computational complexity. The existing results establish a substantial general theory, but they also indicate that exact structure, exact constants, and exact thresholds remain challenging across several of the most natural regimes.