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 $\le r$ vertices has chromatic number $\le 3$, contains an odd cycle with at least $f(r)$ diagonals?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Voss proved (answering a conjecture of Erdős) that **every graph with chromatic number (\ge 4) and with no (K_4) subgraph contains an odd cycle with at least two chords** (i.e., at least two “diagonals” of that cycle). ([ScienceDirect][1]) In particular, if $G$ is (K_4)-free and (\chi(G)=4), then $G$ must contain an od…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.