Mathematicians prove graph sandwich conjecture after two decades

Overview

Updated 3 hours ago

In 2004, two mathematicians hypothesized a way to "sandwich" a hard-to-analyze random graph between two simpler ones. Twenty-two years later, three researchers at the University of Warwick proved it.

The proof connects two of the most studied random graph models, letting mathematicians prove properties of regular graphs using the vast literature on binomial graphs. It's a meta-theorem that unlocks dozens of results at once.

Why it matters

The proof lets mathematicians prove properties of random regular graphs using the larger body of work on binomial graphs, streamlining dozens of proofs.

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Key Indicators

Years from conjecture to proof

Kim and Vu proposed the sandwich conjecture in 2004; the proof was completed in 2025.

Mathematicians who completed the proof

Richard Montgomery, Natalie Behague, and Daniel Iľkovič at the University of Warwick.

Papers on arXiv

The proof was posted to arXiv in October 2025 (paper 2510.20765).

Voices

Curated perspectives — historical figures and your fellow readers.

Ever wondered what historical figures would say about today's headlines?

People Involved

Richard Montgomery

Professor of Mathematics, University of Warwick

Led the team that proved the sandwich conjecture

Natalie Behague

Postdoctoral researcher, University of Warwick

Co-author of the sandwich conjecture proof

Daniel Iľkovič

Doctoral student, University of Warwick

Co-author of the sandwich conjecture proof

Jeong Han Kim

Mathematician, co-conjecturer

Proposed the sandwich conjecture in 2004 while at Microsoft Research

Van Ha Vu

Mathematician, co-conjecturer

Proposed the sandwich conjecture in 2004 while at UC San Diego

Michael Krivelevich

Mathematician, Tel Aviv University

Commented on the proof

Gil Kalai

Mathematician, Hebrew University of Jerusalem

Called the proof a 'meta-theorem'

Organizations Involved

University of Warwick

University

Where the proof was completed

The University of Warwick is where Richard Montgomery, Natalie Behague, and Daniel Iľkovič proved the sandwich conjecture.

Microsoft Research

Research laboratory

Where the conjecture was proposed

Microsoft Research was where Jeong Han Kim worked when he proposed the sandwich conjecture in 2004.

Timeline

January 2004 September 2026

5 events Latest: Today

Tap a bar to jump to that date

September 2026
Quanta Magazine publishes article on the proof
Today Publication

Quanta Magazine publishes a detailed account of the sandwich conjecture proof, bringing the result to a wider audience.
Today
October 2025
Proof posted to arXiv
Publication

The three mathematicians post their proof of the sandwich conjecture to arXiv as paper 2510.20765.
October 1st, 2025
January 2023
Warwick trio begins work on the conjecture
Research

Richard Montgomery, Natalie Behague, and Daniel Iľkovič start thinking about building a random regular graph and a random binomial graph edge by edge.
January 1st, 2023
January 2019
Gao and colleagues prove a related result
Research

A 2019 result by Gao and two colleagues provides a technique that the Warwick team later adapts heavily for the sandwich proof.
January 1st, 2019
January 2004
Kim and Vu propose the sandwich conjecture
Conjecture

Jeong Han Kim and Van Ha Vu hypothesize that a regular graph can be sandwiched between two binomial graphs, connecting two random graph models.
January 1st, 2004

Scenarios

Sandwich proof unlocks new results in graph theory

Likely Resolves by End of 2027

Discussed by: Quanta Magazine, mathematicians in the field

With the sandwich conjecture resolved, mathematicians can draw on the vast literature about binomial graphs to prove properties of regular graphs. The Quanta article notes that new results are already starting to appear. This scenario sees a cascade of new papers using the sandwich theorem.

Sandwich method extended to more complex graph structures

Possible Resolves by End of 2028

Discussed by: Quanta Magazine, Natalie Behague

The article notes that researchers hope to make "even more complicated sandwiches" with alternating layers of binomial and regular graphs, or with other ingredients. This scenario sees the method generalized to new graph classes.

Proof faces peer review scrutiny

Likely Resolves by Q2 2027

Discussed by: Standard academic process

The proof has been posted to arXiv but has not yet been formally peer reviewed at a journal. This scenario sees the paper go through standard review, possibly with minor corrections, and get accepted at a major mathematics journal.

Historical Context

3 moments from history that rhyme with this story — and how they unfolded.

1959

Erdős-Rényi random graph model (1959)

Paul Erdős and Alfréd Rényi introduced the binomial random graph model, where each possible edge between n vertices appears independently with probability p. This became one of the most studied objects in graph theory.

Then

The model became a standard tool for studying network properties.

Now

A vast literature developed on binomial random graphs, which the sandwich conjecture now connects to regular graphs.

Why this matters now

The binomial graph is one half of the sandwich. The conjecture's power comes from connecting this well-studied model to regular graphs.

1975

Szemerédi regularity lemma (1975)

Endre Szemerédi proved a lemma showing that any large graph can be approximated by a union of random-like bipartite graphs. It became a foundational tool in extremal graph theory.

Then

The lemma enabled proofs of many results about dense graphs.

Now

It became a standard meta-tool, similar to how the sandwich theorem now serves as a meta-theorem for regular graphs.

Why this matters now

Both are meta-theorems that let mathematicians prove many results at once by establishing a structural connection between graph classes.

1980

Bollobás's work on random regular graphs (1980)

Béla Bollobás developed techniques for analyzing random regular graphs, where every vertex has the same degree. These graphs are harder to analyze than binomial graphs because edges are not independent.

Then

Bollobás's methods enabled the first systematic study of random regular graphs.

Now

The regular graph model became a standard tool, but proving properties of it required separate work for each property.

Why this matters now

The sandwich conjecture was motivated by the difficulty of proving properties of regular graphs. The proof now lets mathematicians transfer results from binomial to regular graphs.