1. Permutations and Hamiltonian Circuits Larry Griffith Basic Notions Seminar October 10, 2007

2. Warning! These slides are the FORMAL statements I will be doing informal examples and explanations.

3. 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 (e 1, e 2, …, e n ) such that each point of the graph is the endpoint of exactly one edge e j, the endpoint of e j is the starting point of e j+1, and the endpoint of e n is the starting point of e 1.

4. 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

5. Hamiltonian k-paths Let k be an integer between 1 and n. Partition the nodes of G into G 1, …, G k. A Hamiltonian k-path is a collection of Hamiltonian paths for G 1, …, G k. –Examples Alternating theorem –# of 1-paths - # of 2-paths + # of 3-paths … ± # of n-paths = (-1) n+1 (det(adjacency matrix))

6. 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.

7. 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

8. 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

9. 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.

10. 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.