Schur-positivity for generalized nets

TL;DR

This paper introduces a new combinatorial approach using special rim hook G-tabloids to prove Schur-positivity for a family of claw-free graphs called generalized nets, advancing understanding of chromatic symmetric functions.

Contribution

It develops a novel combinatorial method with recurrence relations to establish Schur-positivity for generalized nets and similar claw-free graphs.

Findings

Proves generalized nets are Schur-positive.

Introduces special rim hook G-tabloids for computing Schur coefficients.

Provides a new approach for studying Schur-positivity in graph theory.

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. We construct sign-reversing maps on these special rim hook G-tabloids to obtain a recurrence relation for the Schur coefficients of a family of claw-free graphs called generalized nets, then we prove the entire family is Schur-positive. We subsequently determine an analogous recurrence relation for another, similar family of claw-free graphs. Thus, we demonstrate a new method for proving Schur-positivity of chromatic symmetric functions, which has the potential to be applied to make further progress toward the aforementioned conjecture.