Hamiltonicity of Bell and Stirling Colour Graphs

TL;DR

This paper investigates Hamiltonian properties of Bell and Stirling colour graphs derived from arbitrary graphs and trees, establishing conditions under which these graphs are Hamiltonian, with optimal bounds proven.

Contribution

It proves that most graphs have Hamiltonian $n$-Bell colour graphs and identifies conditions for Hamiltonicity in Stirling colour graphs of trees, extending known results.

Findings

Most graphs have Hamiltonian $n$-Bell colour graphs.

For $k \\geq 4$, Stirling colour graphs of trees are Hamiltonian.

3-Bell colour graphs of trees are Hamiltonian for trees with at least 3 vertices.

Abstract

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.