Hamiltonicity of Bell and Stirling Colour Graphs

Paper

Abstract

For a graph

and a positive integer

, the

-Bell colour graph of

is the graph whose vertices are the partitions of

into at most

independent sets, with two of these being adjacent if there exists a vertex

such that the partitions are identical when restricted to

. The

-Stirling Colour graph of

is defined similarly, but for partitions into exactly

independent sets. We show that every graph on

vertices, except

and

, has a Hamiltonian

-Bell colour graph, and this result is best possible. It is also shown that, for

, the

-Stirling colour graph of a tree with at least

vertices is Hamiltonian, and the 3-Bell colour graph of a tree with at least 3 vertices is Hamiltonian.