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combinatorics / Graph theory - chi-boundedness

Sivaraman's Perfect-Divisibility Characterization Question

Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\chi(H) \le \binom{\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.

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combinatoricsAug 14, 2026Significance 4/100Registry: unreviewed

Sivaraman's Perfect-Divisibility Characterization Question

Prior state unknown→disproved

Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\chi(H) \le \binom{\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.

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Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\chi(H) \le \binom{\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.

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