TJHSST Senior Research Project Investigation of Minimal Conjugate Generators for the Symmet.pdfVIP
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TJHSST Senior Research Project Investigation of Minimal Conjugate Generators for the Symmet
TJHSST Senior Research Project
Investigation of Minimal Conjugate Generators for the Symmetric
Group and their Cayley Graphs
2007-2008
Jacob Steinhardt
September 7, 2007
Abstract
We investigate subsets of the symmetric group with structure similar to that of a graph. The
“trees” of these subsets then lead to minimal highly symmetric generating sets of the symmetric
group. We show that there exist generating sets among these with edge-transitive Cayley graphs
and investigate them in relation to the Lovasz conjecture.
Keywords: Cayley graph, Lovasz conjecture, Hamiltonian cycle, conjugate generators, sym-
metric group, quasi-hamiltonicity.
1 Introduction - Purpose and Scope
Note that a graph can be defined as a collection of vertices and edges. Two vertices are adjacent
if there exists an edge connecting them, and two vertices v1 and v2 are connected if there exists
a sequence of adjacent vertices containing v and v . On the other hand, consider the following
1 2
definition: Given a collection of vertices V and a collection of edges E , we can let each element
of E act on V as a transposition swapping the two vertices on which E is incident. If we then let
multiplication in E extend through the definitions of a group action, E generates a subgroup of the
symmetric group acting on V (we denote this subgroup as E ). Then we say that v1 and v2 are
adjacent if (v v ) ∈ E , and that v and v are connected if (v v ) ∈ E . Additionally, connected
1 2 1
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