Bell coloring graphs: realizability and reconstruction
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912757367963648 |
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| author | Asgarli, Shamil Krehbiel, Sara MacLean, Simon |
| author_facet | Asgarli, Shamil Krehbiel, Sara MacLean, Simon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bell coloring graphs: realizability and reconstruction Asgarli, Shamil Krehbiel, Sara MacLean, Simon Combinatorics Primary 05C15, Secondary 05C60, 05C70, 05C75 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. |
| title | Bell coloring graphs: realizability and reconstruction |
| topic | Combinatorics Primary 05C15, Secondary 05C60, 05C70, 05C75 |
| url | https://arxiv.org/abs/2512.10177 |