Hamiltonian Graphs By: Matt Connor Fall 2013

Hamiltonian Graphs Abstract Algebra Graph Theory Hamiltonian Graphs

•Similar to Koenigsberg •Became a more popular field of study in the mid 1900’s •Became represented as points and lines • More difficult than Eulerian to prove

Sir William Rowan Hamilton Irish Mathematician from the early 1800’s Other contributions include discovery of the Quaternions Hamilton also studied the directionality of graphs.

This type of problem is often referred to as the traveling salesman or postman problem. The idea came from the Icosian game “A traveller wants to visit 20 towns on the vertices of a dodecahedron, going once to every town and returning to the starting point.”

Vertex- a single point on a graph. Edge- connects two vertices. Hamiltonian path-a path going through every vertex of the graph once and only once. Hamiltonian circuit- a closed path going through every vertex of the graph once and only once AND ends at the same vertex it began. Vertex- a single point on a graph. Edge- connects two vertices.

Adjacent vertices- two vertices that share an edge Degree of Vertex- number of connected edges. (denoted deg(v))

Difference Between Path and Circuit

Examples of Hamiltonian Circut

Few theorems about Hamiltonian Circuits •This first theorem to prove that a graph is Hamiltonian is from Dirac in 1952 Theorem: If G is a graph with n vertices, where n≥3 and deg(v)≥n/2, for every vertex v of G, then G is Hamiltonian

This second theorem was produced by Ore in 1960 Theorem: If G is a graph of order n≥3 such that for all distinct non adjacent pairs of vertices u and v, deg(u)+deg(v)≥n, then G is Hamiltonian

•Both of the previous results consider the fact that the more edges a graph has, the more likely it is Hamiltonian. •This just refers to having more opportunities because there are more possible paths. •The more theorems that we look at the more complex they become to confirm a graph.

•Some of the other Theorems include an idea called connectivity. •This is the minimum number of vertices whose removal results in a disconnected graph. •They then use this and relate it to the degree of vertices. This is an example of a 2-connected graph

Graph of Hamilton (dodecahedron) All complete graphs (every vertex connected) Planar 4-connected Platonic solids- regular polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex A few known Hamiltonian Graphs

http://www3.ul.ie/cemtl/pdf%20files/cm2/GraphEulerHamilt on.pdf http://mathafou.free.fr/themes_en/kgham.html http://www.rose- hulman.edu/mathjournal/archives/2000/vol1- n1/paper4/v1n1-4pd.PDF http://www.math.uni- hamburg.de/home/diestel/books/graph.theory/preview/Ch1 0.pdf http://www.britannica.com/EBchecked/topic/253431/Sir- William-Rowan-Hamilton