Permutations and Hamiltonian Circuits

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 (e1, e2, …, en) such that each point of the graph is the endpoint of exactly one edge ej, the endpoint of ej is the starting point of ej+1, and the endpoint of en is the starting point of e1.

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 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))

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.