Labeling Hamiltonian Cycles of the Johnson Graph

George Barnes1, Inessa Levi2
1Professor Emeritus Mathematics Department University of Louisville Louisville KY 40292
2Professor Mathematics Department Columbus State University Columbus, GA 31904

Abstract

A non-empty \( r \)-element subset \( A \) of an \( n \)-element set \( X_n \), and a partition \( \pi \) of \( X_n \), are said to be orthogonal if every class of \( \pi \) meets \( A \) in exactly one element. A partition type is determined by the number of classes of each distinct size of the partition. The Johnson graph \( J(n,r) \) is the graph whose vertices are the \( r \)-element subsets of \( X_n \), with two sets being adjacent if they intersect in \( r-1 \) elements. A partition of a given type \( \tau \) is said to be a \( \tau \)-label for an edge \( AB \) in \( J(n,r) \) if the sets \( A \) and \( B \) are orthogonal to the partition. A cycle \( \mathcal{H} \) in the graph \( J(n,r) \) is said to be \( \tau \)-labeled if for every edge of \( \mathcal{H} \), there exists a \( \tau \)-label, and the \( \tau \)-labels associated with distinct edges are distinct. Labeled Hamiltonian cycles are used to produce minimal generating sets for transformation semigroups. We identify a large class of partition types \( \tau \) with a non-zero gap for which every Hamiltonian cycle in the graph \( J(n,r) \) can be \( \tau \)-labeled, showing, for example, that this class includes all the partition types with at least one class of size larger than 3 or at least three classes of size 3.

Keywords: partition, graph, Hamiltonian cycle, labels. MOS Classification number: 05A18, 05A17, 05C20