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.”
• 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
•Degree
vertices- two vertices that share an edge
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
A few known Hamiltonian Graphs
• Graph
• All
of Hamilton (dodecahedron)
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
•
http://www3.ul.ie/cemtl/pdf%20files/cm2/GraphEulerHamilt
on.pdf
•
http://mathafou.free.fr/themes_en/kgham.html
•
http://www.rosehulman.edu/mathjournal/archives/2000/vol1n1/paper4/v1n1-4pd.PDF
•
http://www.math.unihamburg.de/home/diestel/books/graph.theory/preview/Ch1
0.pdf
•
http://www.britannica.com/EBchecked/topic/253431/SirWilliam-Rowan-Hamilton