Strongly Set-Colorable Graphs: A Complete Characterization

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 $G$ 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.