Reconstructing a graph from its Bell colouring graph

TL;DR

This paper characterizes when two graphs have identical Bell colouring graphs and shows that certain graphs are uniquely determined by these graphs, with implications for graph reconstruction.

Contribution

It provides precise conditions under which graphs are uniquely identified by their Bell colouring graphs and related subgraphs, extending graph reconstruction results.

Findings

Graphs with no vertices of degree n-1 are uniquely determined by their Bell colouring graph.

Graphs with maximum degree less than (n/9 - 1/3) are uniquely identified by their Bell k-colouring graph for k > chromatic number.

Results can be restated in terms of partitions into cliques via graph complements.

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.