Reconstructing a graph from its Bell colouring graph

Abstract

The Bell colouring graph $\mathcal{B}(G)$ of a graph $G$ is the graph whose vertices are the partitions of the vertex set of $G$ into independent sets, with an edge between two partitions if and only if one can be obtained from the other by changing the part of a single vertex of $G$. Given a natural number $k$, the Bell $k$-colouring graph $\mathcal{B}_k(G)$ and the upper-Bell $k$-colouring graph $\mathcal{B}_{\geq k}(G)$ are the induced subgraphs of $\mathcal{B}(G)$ consisting of all partitions with at most $k$ parts and at least $k$ parts, respectively. We determine precisely when two finite graphs have isomorphic Bell colouring graphs. In particular, we show that every $n$-vertex graph $G$ with no vertices of degree $n-1$ is uniquely determined by its Bell colouring graph $\mathcal{B}(G)$, and by its upper-Bell colouring graph $\mathcal{B}_{\geq k}(G)$ if $k\leq n-2$. We also show that every $n$-vertex graph with maximum degree $\Delta(G)< \frac{1}{9}n-\frac{1}{3}$ is uniquely determined by its Bell $k$-colouring graph $\mathcal{B}_k(G)$ if $k>\chi(G)$. By taking graph complements, each of these results can be restated in terms of partitions into cliques.