Two line graphs with 12 vertices have non-Schur-positive chromatic symmetric functions, and a bipartite Schur-positive graph has an unsaturated Newton polytope, disproving two conjectures.
On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture
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abstract
The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic refinement for unit interval graphs. Using the $\mathbb{A}_{q,t}$ algebra, we give an expansion of these $q$-chromatic symmetric functions into Macdonald polynomials. Upon setting $t=1$, we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting $t=0$, we obtain an expansion into Hall-Littlewood symmetric functions.
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Chromatic symmetric functions of claw-free graphs are not Schur positive
Two line graphs with 12 vertices have non-Schur-positive chromatic symmetric functions, and a bipartite Schur-positive graph has an unsaturated Newton polytope, disproving two conjectures.