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Read's conjecture
Mathematical theorem first conjectured by Ronald Read

Read's conjecture is a conjecture, first made by Ronald Read, about the unimodality of the coefficients of chromatic polynomials in the context of graph theory. In 1974, S. G. Hoggar tightened this to the conjecture that the coefficients must be strongly log-concave. Hoggar's version of the conjecture is called the Read–Hoggar conjecture.

The Read–Hoggar conjecture had been unresolved for more than 40 years before June Huh proved it in 2009, during his PhD studies, using methods from algebraic geometry.

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References

  1. Baker, Matthew (January 2018). "Hodge theory in combinatorics". Bulletin of the American Mathematical Society. 55 (1): 57–80. arXiv:1705.07960. doi:10.1090/bull/1599. ISSN 0273-0979. S2CID 51813455. https://www.ams.org/bull/2018-55-01/S0273-0979-2017-01599-6/ ↩

  2. R. C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52–71. MR0224505 (37:104) ↩

  3. Hoggar, S. G (1974-06-01). "Chromatic polynomials and logarithmic concavity". Journal of Combinatorial Theory. Series B. 16 (3): 248–254. doi:10.1016/0095-8956(74)90071-9. ISSN 0095-8956. https://doi.org/10.1016%2F0095-8956%2874%2990071-9 ↩

  4. Huh, June. "Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries" (PDF). https://web.northeastern.edu/martsinkovsky/p/rtrt/20152016/huh-slides.pdf ↩

  5. Baker, Matthew (January 2018). "Hodge theory in combinatorics". Bulletin of the American Mathematical Society. 55 (1): 57–80. arXiv:1705.07960. doi:10.1090/bull/1599. ISSN 0273-0979. S2CID 51813455. https://www.ams.org/bull/2018-55-01/S0273-0979-2017-01599-6/ ↩

  6. "He Dropped Out to Become a Poet. Now He's Won a Fields Medal". Quanta Magazine. 5 July 2022. Retrieved 5 July 2022. https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705 ↩

  7. Kalai, Gil (July 2022). "The Work of June Huh" (PDF). Proceedings of the International Congress of Mathematicians 2022: 1–16., pp. 2–4. https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/laudatio-jh.pdf ↩

  8. Huh, June (2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society. 25 (3): 907–927. arXiv:1008.4749. doi:10.1090/S0894-0347-2012-00731-0. /wiki/June_Huh ↩