The e-positivity conjecture for hat-chains
A hat-chain is the graph constructed by joining the indicated complete graphs and paths, as in the hat-chain construction preceding the conjecture; an expression is e-positive when its chromatic symmetric function has a nonnegative expansion in the elementary symmetric-function basis.
Hat-chain e-positivity conjecture. Every hat-chain is -positive.
Hat-chains form a family of conjoined graphs, and the conjecture asserts a uniform positivity property for their chromatic symmetric functions. The supplied text does not indicate whether this assertion has been proved or disproved.
References
Primary source
E. Y. J. Qi, D. Q. B. Tang and D. G. L. Wang, “Chromatic symmetric functions of conjoined graphs”, arXiv:2406.01418 (2026).
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