2. Eulerian Graphs
An Eulerian graph is a graph containing
an eulerian cycle.
Hamiltonian Graph
A Hamiltonian graph, also called a
Hamilton graph, is a graph possessing a
Hamiltonian cycle. A graph that is not
Hamiltonian is said to be
nonhamiltonian.
3. Planarity
A graph is said to be embedded in
a surface S when it is drawn on S
so that no two edges intersect. A
graph is called planar if it can be
drawn on a plane without
intersecting edges. A graph is
called non-planar if it is not
planar. A graph that is drawn on
the plane without intersecting
edges is called a plane graph
4. Theorem:
A connected graph G is an Euler
graph if and only if all vertices of G
are of even degree.
A connected graph G is Eulerian if
and only if its edge set can be
decom-posed into cycles.
5. A connected graph G is Eulerian if
there is a closed trail which includes
every edge of G, such a trail is called
an Eulerian trail.
Hamiltonian Cycle
A connected graph G is Hamiltonian if
there is a cycle which includes every
vertex of G; such a cycle is called a
Hamiltonian cycle.
6. Matchings
Any set M of independent lines of a graph G is
called a matching of G
Theorem: A matching M in a graph G is a
maximum matching iff G contains no M-
augmenting path
Remark:
1. Kn have perfect matchings if n is even
2. The number of perfect matchings in complete
bipartile graph Kn,n
Hall’s Marriage Theorem:
Let G be a bipartile graph with bipartition (A,B).
Then G has a matching that saturates all the
vertices of A iff for every subset S
of A
SSN )(
