Chromatic symmetric functions of claw-free graphs are not Schur positive

Abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.

6 pages, 2 figures, v2: updated references and added comment at the end of section 1 stating that the counterexamples (Example 2.1 and Example 2.2) to Conjecture 1.1 were discovered independently at approximately the same time by Jitendra Prajapati