Hamiltonicity of Bell and Stirling Colour Graphs

For a graph $G$ and a positive integer $k$, the $k$-Bell colour graph of $G$ is the graph whose vertices are the partitions of $V$ into at most $k$ independent sets, with two of these being adjacent if there exists a vertex $x$ such that the partitions are identical when restricted to $V - {x}$. The $k$-Stirling Colour graph of $G$ is defined similarly, but for partitions into exactly $k$ independent sets. We show that every graph on $n$ vertices, except $K_n$ and $K_n - e$, has a Hamiltonian $n$-Bell colour graph, and this result is best possible. It is also shown that, for $k \geq 4$, the $k$-Stirling colour graph of a tree with at least $k+1$ vertices is Hamiltonian, and the 3-Bell colour graph of a tree with at least 3 vertices is Hamiltonian.

💡 Research Summary

The paper investigates two families of reconfiguration graphs derived from vertex colourings viewed as partitions into independent sets. For a graph G and a positive integer k, the k‑Bell colour graph Bₖ(G) has as vertices all partitions of V(G) into at most k independent sets; two partitions are adjacent if they become identical after deleting a single vertex. The k‑Stirling colour graph Sₖ(G) is the induced subgraph of Bₖ(G) consisting of partitions into exactly k independent sets. These graphs model “small‑change’’ reconfiguration problems and correspond to Gray codes for the underlying combinatorial objects.

The authors first collect known facts about the ordinary k‑colour graph Cₖ(G) (vertices are proper k‑colourings, adjacency means a single vertex changes colour). They note that Cₖ(G) is connected when k ≥ χ(G)+1 and Hamiltonian when k ≥ χ(G)+2. By collapsing colourings that induce the same partition, Bₖ(G) can be seen as a quotient of Cₖ(G); consequently, Bₖ(G) is connected for k ≥ col(G)+1, where col(G) is the colouring number.

A series of structural lemmas are proved. Proposition 2.5 shows that if H is uniquely k‑colourable and disjoint from G, then Bₖ(H∪G) is isomorphic to Cₖ(G). Proposition 2.6 establishes that for the join G∨H, Bₖ(G∨H) ≅ Bₖ(G) □ Bₖ(H), where □ denotes the Cartesian product. Lemma 2.7 proves that the graph K_r □ + K_s (the Cartesian product K_r □ K_s with an extra “clone’’ vertex attached to one vertex and all its neighbours) possesses a Hamilton path starting at the clone and ending at any other vertex. This lemma becomes a key tool for constructing Hamilton cycles in the Bell colour graphs.

The central result (Theorem 3.1) states that for any n‑vertex graph G that is neither the complete graph Kₙ nor Kₙ with a single edge removed, the n‑Bell colour graph Bₙ(G) contains a Hamilton cycle. The proof proceeds by contradiction: assume a minimal counterexample G, pick two non‑adjacent vertices x and y, and let G′ = G – {x, y}. Any colouring of G can be obtained by extending a colouring of G′ in five ways (both new vertices share a colour already used, one gets a new colour, the other shares an old colour, they receive two distinct new colours, or they share a new colour). Analyzing the numbers ℓ₁ and ℓ₂ of available old colours for x and y, the set of extensions of a fixed colouring of G′ induces a subgraph that contains either K_{ℓ₁} □ K_{ℓ₂} or K_{ℓ₁} □ + K_{ℓ₂} as a spanning subgraph. When G′ is complete (or complete minus an edge), Lemma 2.7 guarantees a Hamilton cycle directly.

If G′ is more general, the minimality of G ensures that B_{n‑2}(G′) already has a Hamilton cycle c₁, c₂, …, c_{m′}. For each c_i, the extensions c_i(xy) (both new vertices share a new colour) and c_i(x|y) (they receive distinct new colours) are linked by a Hamilton path P_i inside the subgraph induced by all extensions of c_i. By stitching together the paths P_i in alternating order (even‑indexed paths connected via the xy‑type extensions, odd‑indexed via the x|y‑type extensions) and adding the necessary connecting edges, a Hamilton cycle in Bₙ(G) is assembled. The authors treat the parity of m′ separately and handle the delicate case where both x and y have degree n‑3, showing that this leads to a contradiction with the minimality assumption. Hence no counterexample exists, establishing the theorem.

The result is shown to be best possible. For Kₙ, Bₙ(Kₙ) reduces to a single vertex, and for Kₙ−e it reduces to K₁ or K₂, both non‑Hamiltonian. Moreover, the authors construct a family G_t = K_{2t} minus a perfect matching. For these graphs, B_{t+ℓ}(G_t) corresponds bijectively to binary strings of length t with at most ℓ ones. When ℓ < t, the resulting graph is not Hamiltonian, demonstrating that for some graphs the threshold k₀ at which Bₖ(G) becomes Hamiltonian is exactly |V(G)|.

In Section 4 the paper turns to trees. It proves that for any tree T with at least k+1 vertices, the k‑Stirling colour graph Sₖ(T) is Hamiltonian whenever k ≥ 4. The case k = 3 is handled separately: the 3‑Bell colour graph B₃(T) is Hamiltonian for every tree with at least three vertices. Conversely, the authors exhibit that the 3‑Stirling colour graph of a star with an odd number of leaves fails to be Hamiltonian, showing the bound k ≥ 4 cannot be lowered in general.

Overall, the paper provides a thorough combinatorial analysis of reconfiguration graphs based on independent‑set partitions. By leveraging Cartesian product structures, a novel “clone‑vertex’’ construction, and careful inductive arguments, it establishes sharp Hamiltonicity results for Bell and Stirling colour graphs, extending the theory of graph recolouring and Gray codes.