Erdős Problem #1091

The problem

Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals?

More generally, is there some $f(r)\to \infty$ such that every graph with chromatic number $4$, in which every subgraph on $\leq r$ vertices has chromatic number $\leq 3$, contains an odd cycle with at least $f(r)$ diagonals?

Statement quoted from erdosproblems.com/1091, maintained by Thomas Bloom. Original source: [Er76c,p.4]. Current status there: solved.

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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