Permutations and Hamiltonian Circuits

Larry Griffith

Basic Notions Seminar

October 10, 2007

Warning!

• These slides are the FORMAL statements

• I will be doing informal examples and explanations.

What is a Hamiltonian circuit?

• A directed graph is a set of n points (nodes) and directed lines (edges) connecting the points

• A Hamiltonian circuit (henceforth called a Hamiltonian 1-path) is an ordered n-tuple of edges (e1, e2, …, en) such that each point of the graph is the endpoint of exactly one edge e j, the endpoint of ej is the starting point of ej+1, and the endpoint of en is the starting point of e1.

Hamiltonian Circuits Problem

• Find an efficient algorithm/method to determine if any graph G has a Hamiltonian circuit.– Examples of graphs with and without such

paths

• This is an unsolved problem in general

Hamiltonian k-paths

• Let k be an integer between 1 and n. Partition the nodes of G into G1, …, Gk. A Hamiltonian k-path is a collection of Hamiltonian paths for G1, …, Gk.

– Examples

• Alternating theorem– # of 1-paths - # of 2-paths + # of 3-paths … ±

# of n-paths = (-1)n+1 (det(adjacency matrix))

Algebraic connection

• Determinants can be efficiently calculated, so this may be a starting point.

• It is difficult to distinguish 1-paths from other k-paths algebraically, which makes it difficult to use this formula.

• One possible way to deal with this is to look at permutation groups.

Hamiltonian k-paths as permutations

• Number the points in some arbitrary fashion

• Path i1 -> i2 -> … -> in as permutation (i1 i2 … in) cycle– k-path becomes permutation with multiple

cycles– Examples– It is still difficult to calculate whether a graph

has a cycle involving all n points

Another way to get permutations

• The numbering was arbitrary.

• Renumbering the points is a permutation.

• Group action– Renumbering the points changes the paths,

i.e. renumbering permuations act on Hamiltonian k-path permutations

Central point

• Renumbering permutations change Hamiltonian k-paths permutations into other k-paths with matching lengths

• All k-paths with a fixed set of lengths can be obtained from a particular one by renumbering

• “Orbits of the group action” are characterized by k integers whose sum is n.

Characterizing 1-paths

• Orbits of 1-paths are not groups, i.e. you can “multiply” or compose permutations in them and get permutations that are not 1-paths.

• They generate either the group of all permutations (if n is odd) or the group of “even” permutations (if n is even)

• All other orbits generate smaller groups.