A proof of the Murty--Simon conjecture on diameter-$2$-critical graphs

Sanjana Das; Jacob Fox

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arXiv · 2610.39032

A proof of the Murty--Simon conjecture on diameter-$2$-critical graphs

Sanjana Das·Jacob Fox

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Abstract

The Murty-Simon conjecture states that every $n$-vertex diameter-$2$-critical graph has at most $\lfloor n^2/4 \rfloor$ edges, with equality only for the balanced complete bipartite graph. Using regularity methods, Füredi proved this for all sufficiently large $n$ (specifically, $n$ greater than a tower of $2$'s of height roughly $10^{14}$). We present a proof of the Murty-Simon conjecture that works for all $n$. A proof of the $\lfloor n^2/4 \rfloor$ bound was originally found by AI; we simplified this proof and extended its ideas to get a proof of the full conjecture.

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Sanjana Das, Jacob Fox. 2026-10-28. A proof of the Murty--Simon conjecture on diameter-$2$-critical graphs. https://arxiv.org/abs/2610.39032

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