A Schur-positivity classification for complete multipartite graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910176106250240 |
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| author | Shelburne, Ethan van Willigenburg, Stephanie |
| author_facet | Shelburne, Ethan van Willigenburg, Stephanie |
| contents | 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 $K_λ$ is Schur-positive if and only if either $λ_i\in \{1,2\}$ for all $i$ or $λ=(3,2^β)$ for some $β\ge 1$. 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 $K_{(3,2^β)}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26158 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Schur-positivity classification for complete multipartite graphs Shelburne, Ethan van Willigenburg, Stephanie Combinatorics 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 $K_λ$ is Schur-positive if and only if either $λ_i\in \{1,2\}$ for all $i$ or $λ=(3,2^β)$ for some $β\ge 1$. 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 $K_{(3,2^β)}$ 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. |
| title | A Schur-positivity classification for complete multipartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.26158 |