A Schur-positivity classification for complete multipartite graphs

TL;DR

This paper classifies exactly which complete multipartite graphs have non-negative Schur function expansions of their chromatic symmetric functions, extending previous bipartite and tripartite results.

Contribution

It provides a complete classification of Schur-positivity for all complete multipartite graphs, including new combinatorial formulas for Schur coefficients.

Findings

Complete multipartite graphs are Schur-positive iff all parts are 1 or 2, or the graph is of form (3, 2^β).

Extended classification from bipartite and tripartite to all multipartite graphs.

Derived a simplified formula for Schur coefficients of incomparability graphs.

Abstract

A graph is Schur-positive if its chromatic symmetric function expands non-negatively in the Schur basis. We determine a full Schur-positivity classification for complete multipartite graphs by showing that a complete multipartite graph $K_\lambda$ is Schur-positive if and only if either $\lambda_i\in \{1,2\}$ for all $i$ or $\lambda=(3,2^\beta)$ for some $\beta\ge 1$. These results extend earlier classifications for complete bipartite and complete tripartite graphs to full generality. Our proofs combine structural arguments ruling out most cases, with a combinatorial analysis of Schur coefficients for the remaining family $K_{(3,2^\beta)}$ via special rim hook $G$-tabloids. Along the way, we establish a simpler formula for Schur coefficients of incomparability graphs, which we then apply to compute the coefficients of interest in terms of non-increasing sequences.