Bell coloring graphs: realizability and reconstruction

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.