Open Collections

Abstract

A graph is Schur-positive if its chromatic symmetric function expands nonnegatively in the Schur basis. All claw-free graphs are conjectured to be Schur-positive. We introduce a combinatorial object corresponding to a graph G, called a special rim hook G-tabloid, which is a variation on the special rim hook tabloid. These objects can be employed to compute any Schur coefficient of the chromatic symmetric function of a graph. Special rim hook tabloids have previously been used to prove the non-Schur-positivity of some graphs. We construct sign-reversing maps on these special rim hook G-tabloids to prove that a family of claw-free graphs called generalized nets are Schur-positive. Thus, we demonstrate a new method for proving the Schur-positivity of graphs, which has the potential to be applied to make further progress toward the aforementioned conjecture.