This document discusses several graph theory concepts: - An Eulerian graph contains an Eulerian cycle that uses each edge exactly once. A Hamiltonian graph contains a Hamiltonian cycle that visits each vertex exactly once. - A planar graph can be drawn in a plane without edge crossings, while a non-planar graph cannot be drawn this way. - A graph is Eulerian if all vertices have even degree or if its edges can be decomposed into cycles. A graph is Hamiltonian if it contains a cycle visiting each vertex. - A matching in a graph is a set of independent edges, and a maximum matching uses the most possible edges. Hall's Marriage Theorem gives conditions for a bipartite graph to have a matching

1. GRAPH THEORY 15UMTC53
III B.Sc MATHMATICS A & B
Mrs. G.Nagalakshmi
Mrs. K. Muthulakshmi

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

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 )(
SSN )(