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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2508.13210 |
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| _version_ | 1866915450161463296 |
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| author | Abhishek, Kumar |
| author_facet | Abhishek, Kumar |
| contents | In this note, we revisit the notion of strong set-colorings introduced by Hegde (2009) and completed by equivalences due to Boutin et al. (2010) and provide a necessary and sufficient \emph{Steiner packing} characterisation: a finite graph $G$ is strongly set-colorable if and only if its associated $3$-uniform hypergraph is a $(2,3,2^{n}-1)$-packing of the unique Steiner triple system $S(2,3,2^{n}-1)$. This unification allows many earlier necessary conditions to be derived instantly as corollaries, streamlining the structure theory of strongly set-colorable graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13210 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly Set-Colorable Graphs: A Complete Characterization Abhishek, Kumar Combinatorics 05C65, 05C78 G.2 In this note, we revisit the notion of strong set-colorings introduced by Hegde (2009) and completed by equivalences due to Boutin et al. (2010) and provide a necessary and sufficient \emph{Steiner packing} characterisation: a finite graph $G$ is strongly set-colorable if and only if its associated $3$-uniform hypergraph is a $(2,3,2^{n}-1)$-packing of the unique Steiner triple system $S(2,3,2^{n}-1)$. This unification allows many earlier necessary conditions to be derived instantly as corollaries, streamlining the structure theory of strongly set-colorable graphs. |
| title | Strongly Set-Colorable Graphs: A Complete Characterization |
| topic | Combinatorics 05C65, 05C78 G.2 |
| url | https://arxiv.org/abs/2508.13210 |