Two infinite families of counterexamples to the Stanley--Gasharov conjecture
The Stanley--Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley--Gasharov conjecture, thereby answering a question of Matherne and Morales. Every graph in the first family is a line graph, whereas no graph in the second family is a line graph. Prajapati further showed that the graph $G_2$, which has $12$ vertices and $21$ edges, is the smallest counterexample under the ordering that first compares the numbers of vertices and then the numbers of edges. We show that $G_2$ is also the smallest counterexample under the reverse ordering, which first compares the edge numbers and then the vertex numbers. Similarly, we exhibit a graph $Q$ with $13$ vertices and $27$ edges and show that $Q$ is the smallest counterexample that is not a line graph under each ordering. Our two infinite families are obtained from $G_2$ and $Q$, respectively, by adjoining a clique of order at least $4$ and connecting one of its vertices to a distinguished vertex of the original graph by a single edge.