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arXiv · 2508.13210

Strongly Set-Colorable Graphs: A Complete Characterization

Kumar Abhishek
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Abstract

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.

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Kumar Abhishek. 2025-08-16. Strongly Set-Colorable Graphs: A Complete Characterization. https://arxiv.org/abs/2508.13210

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