Erdős Problem #1032

The problem

We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.

Is there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\gg n$?

Statement quoted from erdosproblems.com/1032, maintained by Thomas Bloom. Original source: [Er93,p.341][Va99,3.60]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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