Bell coloring graphs: realizability and reconstruction

TL;DR

This paper studies the structure of Bell coloring graphs, classifies their cliques, characterizes which graphs can be represented as Bell coloring graphs, and proves reconstruction results for trees and general graphs.

Contribution

It provides a structural classification of cliques in Bell coloring graphs, characterizes which graphs are Bell coloring graphs, and establishes reconstruction invariants for trees and general graphs.

Findings

All trees and cycles are Bell coloring graphs.

$K_4-e$ is not a Bell coloring graph.

Bell 3-coloring graphs uniquely identify trees.

Abstract

Given a graph $G$, the Bell $k$-coloring graph $\mathcal{B}_k(G)$ has vertices given by partitions of $V(G)$ into $k$ independent sets (allowing empty parts), with two partitions adjacent if they differ only in the placement of a single vertex. We first give a structural classification of cliques in Bell coloring graphs. We then show that all trees and all cycles arise as Bell coloring graphs, while $K_4-e$ is not a Bell coloring graph and, more generally, $K_n-e$ is not an induced subgraph of any Bell coloring graph whenever $n \geq 6$. We also prove two reconstruction results: the Bell $3$-coloring graph is a complete invariant for trees, and the Bell $n$-coloring multigraph determines any graph up to universal vertices.