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arXiv · 2604.26158

A Schur-positivity classification for complete multipartite graphs

Ethan Shelburne·Stephanie van Willigenburg
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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.

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Ethan Shelburne, Stephanie van Willigenburg. 2026-04-28. A Schur-positivity classification for complete multipartite graphs. https://arxiv.org/abs/2604.26158

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