The chromatic number of heptagraphs

A hole is an induced cycle of length at least 4. A graph is called a pentagraph if it has no cycles of length 3 or 4 and has no holes of odd length at least 7, and is called a heptagraph if it has no cycles of length less than 7 and has no holes of odd length at least 9. Let $ł\ge 2$ be an integer. The current authors proved that a graph is 4- colorable if it has no cycles of length less than $2ł+1$ and has no holes of odd length at least $2ł+3$. Confirming a conjecture of Plummer and Zha, Chudnovsky and Seymour proved that every pentagraph is 3-colorable. Following their idea, we show that every heptagraph is 3-colorable.

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