An infinite family of counterexamples to the Stanley--Gasharov conjecture

The Stanley--Gasharov conjecture asserts that the chromatic symmetric function of every claw-free graph is Schur-positive.

Prajapati, and independently Matherne and Morales, found counterexamples, and the latter asked for an infinite family of counterexamples.

Combining Prajapati's complete census through order~$12$ with an exact census of the $144{,}492$ previously untreated connected claw-free graphs on~$n$ vertices and~$m$ edges with $13\le n\le 21$ and $n-1\le m\le 20$, we show that their counterexample graph~$G_2$ with $12$ vertices and $21$ edges is the unique minimum counterexample under the edge-first, vertex-second order.

We also construct an infinite family of connected line graphs whose chromatic symmetric functions are not Schur-positive.

This yields an infinite family of counterexamples to the Stanley--Gasharov conjecture.