Semirandom Graph Process

• Semirandom Graph Process is an online model where each round reveals a random vertex and the player adaptively selects a partner vertex to add an edge.
• It simulates classical random graph models such as G(n,m) and k-out through coupling strategies, enabling precise analysis of thresholds for connectivity and subgraph properties.
• The process underpins optimal round complexity results and supports various extensions, including hypergraph generalizations, no-replacement, and pre-positional variants.

The semirandom graph process is a one-player online graph-building model on vertex set $[n]=\{1,\dots,n\}$ in which each round exposes one uniformly random vertex and the player adaptively chooses the other endpoint of the new edge. Starting from the empty graph $G_0$, round $t$ reveals $u_t\in[n]$, the player chooses $v_t\in[n]$ using the full history, and the edge $u_tv_t$ is added. In the standard formalization, loops and parallel edges are allowed, so the evolving object is naturally a multigraph, although most target properties are interpreted on the underlying simple graph. The central question is a hitting-time problem for monotone graph properties: how many rounds are necessary and sufficient, with high probability, for an optimal strategy to force a property $\mathcal P$ to appear (Ben-Eliezer et al., 2018, Behague et al., 2021).

1. Formal definition and threshold viewpoint

The foundational model introduced in “Semi-random graph process” (Ben-Eliezer et al., 2018) fixes an empty graph on $[n]$ and repeats a two-stage move: a vertex is offered uniformly at random, then Builder chooses an additional vertex and adds the incident edge. In the terminology used in later work, the offered endpoint is the “square” and the chosen endpoint is the “circle,” so each round may also be viewed as adding an oriented arc from the random vertex to the strategic one (Behague et al., 2021, Gamarnik et al., 2023).

The target of the process is usually a monotone increasing graph property. For a strategy $S$, one studies the smallest number of rounds needed so that the property holds with probability at least $p$; optimizing over all strategies yields the process threshold for that property (Ben-Eliezer et al., 2018). Later papers also use asymptotic threshold notation such as $G_0$0 for the round scale at which success becomes likely, and in the $G_0$1-choice extension $G_0$2 for the normalized threshold under $G_0$3 offered random vertices per round (Behague et al., 2021, Prałat et al., 2023).

A standard distinction is between the online model, where the strategy reacts only to the current offer and the past, and an offline model, where the full future offer sequence is known in advance. That distinction matters sharply for fixed-subgraph embedding: the offline problem reduces to an occupancy condition on offer counts, whereas the online problem requires genuinely adaptive control (Ben-Eliezer et al., 2018).

2. Foundational results and simulation of classical models

A basic feature of the semirandom graph process is that it can realize or approximate several classical random graph models by suitable strategies. There is a strategy $G_0$4 whose distribution is exactly the random multigraph process $G_0$5. There is also a strategy $G_0$6 such that when $G_0$7, the semirandom process after $G_0$8 rounds can be coupled to $G_0$9 with containment $t$0 with high probability, and when $t$1 the coupling is exact with high probability. For fixed $t$2, there is a strategy $t$3 coupling the process to the $t$4-out model within $t$5 rounds, and there is likewise a coupling to the classical minimum-degree process with only $t$6 multiplicative overhead (Ben-Eliezer et al., 2018).

These couplings explain why many upper bounds in the semirandom literature are obtained by importing known behavior from $t$7, $t$8-out graphs, or minimum-degree processes. In particular, the original paper derived online thresholds for minimum degree and $t$9-connectivity from comparison with the min-degree process, while later work used $u_t\in[n]$0-out and $u_t\in[n]$1-out couplings as crude upper bounds for perfect matching and Hamiltonicity (Ben-Eliezer et al., 2018, Prałat et al., 2023).

The offline theory is especially clean for fixed subgraphs. If $u_t\in[n]$2 is fixed and $u_t\in[n]$3 is the minimum possible maximum out-degree over all orientations of $u_t\in[n]$4, then the offline threshold for containing $u_t\in[n]$5 satisfies

$u_t\in[n]$6

with high probability. The same paper gives an offline minimum-degree-$u_t\in[n]$7 threshold $u_t\in[n]$8, where $u_t\in[n]$9 is the unique positive root of an explicit equation, and notes that $v_t\in[n]$0 as $v_t\in[n]$1 (Ben-Eliezer et al., 2018).

3. Fixed subgraphs and the degeneracy principle

The central structural theorem for the online process concerns forcing a fixed graph $v_t\in[n]$2 as a subgraph. Writing

$v_t\in[n]$3

the key parameter is the degeneracy of $v_t\in[n]$4: $v_t\in[n]$5 is $v_t\in[n]$6-degenerate if every subgraph has a vertex of degree at most $v_t\in[n]$7, equivalently if its vertices can be ordered so that each vertex has at most $v_t\in[n]$8 earlier neighbors (Behague et al., 2021).

Earlier work proved that if $v_t\in[n]$9 has degeneracy $u_tv_t$0, then there is a strategy succeeding asymptotically almost surely in

$u_tv_t$1

rounds for any diverging $u_tv_t$2, and also showed sharpness for $u_tv_t$3. “Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs” (Behague et al., 2021) proves the full conjecture: for every fixed graph $u_tv_t$4 of degeneracy $u_tv_t$5, no strategy succeeds asymptotically almost surely when $u_tv_t$6, while some strategy succeeds asymptotically almost surely whenever $u_tv_t$7. Thus the threshold is exactly

$u_tv_t$8

For $u_tv_t$9, that is, for forests, the threshold is trivial: $\mathcal P$0 (Behague et al., 2021).

The lower bound is one of the main technical contributions of that paper. The argument passes to the nonempty $\mathcal P$1-core $\mathcal P$2 of $\mathcal P$3, fixes an orientation and edge ordering, and defines a recursive weight $\mathcal P$4 on vertices. Every vertex in a minimum-degree-$\mathcal P$5 graph is shown to have weight at least $\mathcal P$6, and the number of possible images of a vertex $\mathcal P$7 in $\mathcal P$8 is bounded by order

$\mathcal P$9

Hence, when $[n]$0, a first-moment argument rules out copies of $[n]$1, and therefore of $[n]$2, with high probability (Behague et al., 2021).

The upper bound is constructive and uses a degeneracy ordering. Builder embeds $[n]$3 one vertex at a time, in phases, waiting until the random offers hit vertices often enough to connect a new vertex to its already embedded earlier neighbors. A counting lemma on vertices receiving exactly $[n]$4 random selections supplies the necessary abundance of “good” offers once $[n]$5 (Behague et al., 2021).

This degeneracy principle reappears elsewhere in the literature. The comparison between pre-positional and post-positional variants shows that both have the same asymptotic threshold $[n]$6 for fixed $[n]$7-degenerate subgraphs, and the paper also derives an immediate minor consequence: if a fixed graph contains a cycle, then forcing an $[n]$8-minor has threshold $[n]$9 in both variants (Gao et al., 2023).

4. Spanning properties and optimal round complexity

For spanning properties, the natural scale is linear in $S$0. In the original process, the online threshold for minimum degree at least $S$1 is $S$2, where the constants are inherited from the min-degree process; the paper lists

$S$3

For every fixed $S$4, the threshold for $S$5-vertex-connectivity is asymptotically the same: $S$6 with high probability (Ben-Eliezer et al., 2018).

The case $S$7, left open in the original process, was settled in “On the pre- and post-positional semi-random graph processes” (Gao et al., 2023). That paper proves that both the original post-positional process and Wormald’s pre-positional variant satisfy

$S$8

The upper bound is obtained by first reaching minimum degree $S$9, then connecting the graph and eliminating cut-vertices in $p$0 extra rounds via block-structure analysis (Gao et al., 2023).

Perfect matchings and Hamiltonian cycles have been analyzed most extensively in the $p$1-choice extension, but the $p$2 case is already informative. In the original process, simple couplings to $p$3-out and $p$4-out yield upper bounds $p$5 and $p$6, while fully adaptive augmenting-path and path-extension strategies improve substantially on these couplings. The $p$7-choice paper formulates lower and upper bounds through differential-equation systems and emphasizes that stronger lower and upper bounds are known in the $p$8 case from other papers, although those arguments were not extended there to general $p$9 (Prałat et al., 2023).

A more recent line concerns forcing an arbitrary bounded-degree spanning graph. A prior result of Ben-Eliezer, Gishboliner, Hefetz and Krivelevich gave $G_0$00 rounds in general, and $G_0$01 when $G_0$02. “Optimally building spanning graphs in semirandom graph processes” (Anastos et al., 30 Sep 2025) improves this to

$G_0$03

for every $G_0$04-vertex graph $G_0$05 with maximum degree $G_0$06, which matches the trivial lower bound up to $G_0$07. The proof uses a balanced orientation, an initial greedy phase, a bridging phase based on candidate lists and asymmetric repair, and a final iterative repair phase (Anastos et al., 30 Sep 2025).

5. Major variants of the process

A natural strengthening presents $G_0$08 independent uniformly random vertices in each round, after which the player chooses one offered vertex and any partner. This $G_0$09-process is monotone in $G_0$10: larger $G_0$11 can only help, because extra offered vertices may be ignored. For minimum degree at least $G_0$12, the greedy strategy that always chooses a minimum-degree offered square and joins it to a minimum-degree vertex is proved optimal. The normalized hitting time is a constant $G_0$13 derived from a phase-based ODE system, with

$G_0$14

and the lower bound becomes asymptotically tight either for fixed $G_0$15 and $G_0$16 or for fixed $G_0$17 and $G_0$18. For perfect matching and Hamiltonicity, the same paper gives ODE-based upper and lower bounds and shows that, as $G_0$19,

$G_0$20

The interpretation given there is that multiple choices make it increasingly unlikely that all offered squares miss the useful part of the state space (Prałat et al., 2023).

A different modification is the no-replacement model, where offers arrive in blocks generated by independent random permutations. Within each block, offered vertices do not repeat. This change alters several thresholds. Orientation-based bounds show that if a graph $G_0$21 admits an orientation of maximum out-degree $G_0$22, then $G_0$23, while unavoidable out-degree obstructions force $G_0$24. For minimum degree, the no-replacement thresholds are exact for even degrees and asymptotically explicit for odd degrees: $G_0$25 The same paper shows that perfect matchings typically require $G_0$26 rounds and introduces an urn model that explains the characteristic $G_0$27 threshold for eliminating isolated vertices (Gilboa et al., 2020).

The pre-positional variant reverses the order of moves: the strategic vertex is chosen first and the random endpoint second. For fixed $G_0$28-degenerate subgraphs and fixed $G_0$29-connectedness, the pre- and post-positional processes are asymptotically equally fast, but they are not equivalent for all properties. In particular, there exists a multigraph property—containing an induced simple $G_0$30-cycle—for which the post-positional process succeeds in $G_0$31 rounds while the pre-positional process asymptotically almost surely cannot succeed at all. The same paper conjectures a genuine gap for sufficiently dense bipartite-subgraph forcing (Gao et al., 2023).

Another extension replaces the random star offered each round by a uniformly random spanning tree. In the uniform-spanning-tree semi-random model, Builder chooses one edge from the offered tree. This retains the same exposure-and-choice architecture but reveals a much larger random object each round. The resulting hitting-time theory is near-optimal for several spanning properties: Hamiltonicity can be forced in $G_0$32 rounds, minimum degree $G_0$33 in roughly $G_0$34 rounds, $G_0$35-connectivity in roughly $G_0$36 rounds up to lower-order error, and several $G_0$37-factor results are available for fixed tree-like $G_0$38 (Burova et al., 2022). The optimal $G_0$39 bounded-degree construction theorem of (Anastos et al., 30 Sep 2025) extends to this tree process as well.

6. Hypergraph extensions and terminological boundaries

The semirandom graph process has a direct hypergraph generalization. In the $G_0$40-process, a uniformly random set $G_0$41 of $G_0$42 vertices is revealed and the player chooses $G_0$43 additional vertices to form an $G_0$44-edge. When $G_0$45 and $G_0$46, this is the original graph process. For $G_0$47, the threshold for building a fixed $G_0$48-uniform hypergraph $G_0$49 is again determined solely by degeneracy: $G_0$50 where $G_0$51 is the hypergraph degeneracy. The lower-bound proof adapts the graph argument by giving each hyperedge a leading vertex and working with an auxiliary directed graph; the extra factor of $G_0$52 changes only constants, not the exponent (Behague et al., 2021).

The case $G_0$53 is qualitatively different. “Perfect matchings and loose Hamilton cycles in the semirandom hypergraph model” (Molloy et al., 2023) studies the $G_0$54-offer $G_0$55-uniform model, where each round presents two random vertices and the player chooses a third. That paper proves linear-time asymptotically almost sure construction of both perfect matchings and loose Hamilton cycles, and extends both results to $G_0$56-uniform hypergraphs with the appropriate divisibility conditions. Its main methodological point is that graph-style direct patching fails: in a $G_0$57-uniform process, forcing a prescribed hyperedge requires seeing two of its three vertices together, an event of probability only $G_0$58 in one round. The paper therefore uses a large absorber together with an auxiliary bipartite graph from which it extracts a uniform $G_0$59-out subgraph, taking $G_0$60, and then finishes via Hall-type matching arguments (Molloy et al., 2023).

The phrase “semirandom” is also used in several adjacent literatures, and those models should not be conflated with the semirandom graph process itself. In semirandom stochastic block models, a monotone adversary may add internal cluster edges or increase within-cluster edge probabilities; this is a robustness test for clustering algorithms, not an online edge-building game (Bhaskara et al., 2024). In semirandom hypergraph certification, a random $G_0$61-uniform hypergraph may be strengthened by a monotone adversary who adds edges, and Sum-of-Squares certificates are designed to remain valid under such edge additions (Kothari et al., 9 Mar 2026). In semirandom Hamiltonian $G_0$62-XOR, the support hypergraph may be adversarial while the signs are random (Kocurek, 4 Nov 2025). Even within probabilistic combinatorics, some constrained random graph evolutions are called semirandom in a broader sense—for example the diamond-free process, which repeatedly adds a uniformly random open edge subject to avoiding $G_0$63 (Picollelli, 2010). A common misconception is therefore terminological: “semirandom graph process” in the strict combinatorial sense denotes the offer-and-choice process introduced in (Ben-Eliezer et al., 2018), whereas many other semirandom models impose monotone adversaries or forbidden-configuration constraints rather than adaptive endpoint choice.