Strongly Set-Colorable Graphs: A Complete Characterization
Kumar Abhishek

TL;DR
This paper provides a complete characterization of strongly set-colorable graphs through a Steiner packing framework, unifying previous conditions and advancing the structural understanding of these graphs.
Contribution
It introduces a Steiner packing characterization that fully describes strongly set-colorable graphs, unifying earlier results and simplifying their structural analysis.
Findings
Characterization of strongly set-colorable graphs via Steiner packings.
Unification of previous necessary conditions as corollaries.
Streamlined structure theory for strongly set-colorable graphs.
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 is strongly set-colorable if and only if its associated -uniform hypergraph is a -packing of the unique Steiner triple system . 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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