The Mathematics of
Graphs
Outline
1.
Graphs and Euler Circuits
2.
Weighted Graphs
3.
Planarity and Euler’s Formula
4.
Graph Coloring
Graphs and Euler Circuits
Graph
A graph is a set of points called vertices and line
segments or curves called edges that connects
vertices.
Example: Graph
• Each dot represents a person, and a line segment
connecting two dots.
• It means that those two people are friends on
Facebook.
Terminologies
•
In general, graphs can contain vertices that are not
connected to any edges.
•
If two or more edges connect the same vertices, they
are called multiple edges.
•
If an edge begins and ends at the same vertex, it is
called a loop.
•
A graph is connected if any vertex can be reached from
any other vertex by tracing along edges.
•
A connected graph in which every possible edges is
drawn between vertices is called complete graph.
A
B
vertices not
connected
A
B
Multiple edges
A
loop
Complete Graph
Terminologies
•
A graph is simple if it has no loops and there is at most
one edge between any two vertices in the graph.
Equivalent Graphs
Two or more are called equivalent graphs if the edges
form the same connections of vertices in each graphs.
Euler Circuits
•A
path in a graph can be thought of as a movement from
one vertex to another by traversing edges.
•If a path ends at the same vertex at which it started, it is
considered a closed path, or circuit.
•A circuit that uses every edge, but never uses the same
edge twice, is called an Euler circuit.
•The number of edges that meet at a vertex is called the
degree of a vertex.
A-D-F-G-D-B-E-G-H-E-C-B-A
A-B-C-E-B-D-F-G-H-E-G-D-A
Eulerian Graph Theorem
A connected graph is Eulerian
if and only if
each vertex of the graph is of even degree.
Euler Path
•
An Euler path is a path that uses every edge of a graph
exactly once.
•
An Euler circuit is a circuit that uses every edge of a
graph exactly once.
•
An Euler path starts and ends at different vertices.
•
An Euler circuit starts and ends at the same vertex.
Euler Path Theorem
A connected graph contains an Euler path if and only if the
graph has two vertices of odd degree with all other
vertices of even degree.
Furthermore, every Euler path must start at one of the
vertices of odd degree and end at the other.
1
2
3
4
3-2-4-2-1-3-4
Weighted Graphs
Hamiltonian Circuits
•
A Hamiltonian Circuit is a path that uses each vertex of
a graph exactly once.
•
A graph that contains a Hamiltonian circuit is called
Hamiltonian.
Example: Hamiltonian Circuit
Find a Hamiltonian circuit from this graph.
1-5-4-3-2-12-11-10-9-8-7-6-15-16-17-18-19-20-13-14-1
Example: Hamiltonian Circuit
Find a Hamiltonian circuit from this graph.
1-5-4-8-7-6-15-14-13-12-11-19-20-16-17-18-9-10-3-2-1
Dirac’s Theorem
Consider a connected graph with at least three vertices and
no multiple edges. Let 𝑛 be the number of vertices in the
𝑛
graph. If every vertex has a degree of at least , then the
2
graph must be Hamiltonian.
Dirac’s Theorem
Note:
If the graph does not meet the requirements of the
Dirac’s Theorem, it still might be Hamitonian.
Weighted Graph
A weighted graph is a graph in which each edge
is associated with a value, called a weight.
The Greedy Algorithm
A method of finding a Hamiltonian circuit in a
complete weighted graph is given by the following
greedy algorithm.
1. Choose a vertex to start at, then travel along the
connected edge that has the smallest weight.
2. After arriving at the next vertex, travel along
the edge of smallest weight that connects to a
vertex not yet visited. Continue this process
until you have visited all vertices.
3. Return to the starting vertex.
The Greedy Algorithm
Note:
The greedy algorithm attempts to give a
circuit of minimal total weight, although it
does not always succeed.
Example: Greedy Algorithm
Use the greedy algorithm to find a Hamiltonian circuit
starting at vertex A in the given weighted graph.
A-D-B-C-F-E-A
1-2-3-6-4-5=21
The Edge-Picking Algorithm
Another method of finding a Hamiltonian circuit in a
complete weighted graph is given by the following
edge-picking algorithm.
1. Mark the edge of smallest weight in the graph.
2. Mark the edge of the next smallest weight in the
graph, as long as it does not complete a circuit
and does not add a third marked edge to a single
vertex.
3. Continue the process until you can no longer mark
any edges. Then mark the final edge that
completes the Hamiltonian circuit.
The Edge-Picking Algorithm
Note:
The edge-picking algorithm attempts to give
a circuit of minimal total weight, although it
does not always succeed.
Example: Edge Picking Algorithm
Use the edge-picky algorithm to find a Hamiltonian circuit
in the given weighted graph.
A-D-B-F-E-C-A
1-2-4-4-7-2=20
Planarity and Euler’s Formula
Planar Graphs
A planar graph is a graph that can be drawn so that no
edges intersect each other (except at vertices).
Platonic Solids
Subgraphs
A part of a graph G is called a subgraph of G.
Subgraph Theorem
“If a graph G has a subgraph that is not planar, then G is
also not planar. In particular, if it contains the Utilities
Graph or K5 as a subgraph, G is not planar.”
Nonplanar Graph Theorem
A graph is nonplanar if and only if it has the Utilities Graph
or K5 as a subgraph, or it has a subgraph that can be
contracted to the Utilities Graph or K5.

H is a subgraph of G if

𝑉 𝐻 ⊂𝑉 𝐺

𝐸 𝐻 ⊂𝐸 𝐺
Euler’s Formula
In a connected planar graph drawn with no
intersecting edges, let v be the number of vertices, e
the number of edges, and f the number of faces.
Then v + f = e + 2.
Euler’s Formula
v=6 e=9
v+f=e+2
6+5=9+2
f=5
Graph Colorings
Coloring Map
If the map is divided into regions in some manner, what is the
minimum number of colors required if the neighboring regions are
to be colored differently?
There is a connection between map coloring and graph theory.
Maps can be modeled by graphs using the countries as the vertices
and two vertices (countries) are adjacent if they share a common
boundary.
In graph coloring, each vertex of a graph will be assigned one color
in such away that no two adjacent vertices have the same color.
The interesting idea here is to determine the minimum number of
(distinct) colors to be used so that we can color each vertex of a
graph with no two adjacent vertices have the same color.
Four-Color Theorem
Every planar graph is 4-colourable.
The Chromatic Number of a Graph
The minimum number of colors needed to color a graph so
that no edge connects vertices of the same color is called
the chromatic number.
2-Colorable Graph Theorem
A graph is 2-colourable if and only if it has no circuits that
consist of an odd number of vertices.
Example:
Look at the map. Assume that no
country is split into more than one
piece and countries that touch each
other at just a corner point will not be
considered neighbors.
How many colors are needed?
Example:
Determine whether the graph is 2colorable.
References:
•
Auffmann et al, Mathematical Excursions (2018)
•
https://www.youtube.com/watch?v=nZwSo4vfw6c
•
•
Mame, Neil (Batangas State University), Manalang, Rodman (UE Manila),
Maquiling Rene (Xavier University), Mocorro, Ronald (Leyte Normal University) –
Powerpoint Presentation (2017)
Manalang, R., Powerpoint – University of the East (2017)