Group 9 Research Report 1

7/27/2019 Group 9 Research Report 1

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Research Report 1Problem No 09 : T Graphs

Group 09

7/31/2013

H. P. Hewamalage (110220J)

S. Shobiga (110550U)

W. N. T. Wajirasena (110607D)

S. R. B. V. T. Weerasiri (110615B)

R. A. A. V. A. Wijesundara (110644L)


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Table of ContentsDefinitions Related to T Graphs .................................................................................... 01

Problem 09T Graphs .................................................................................................. 03

Problems Related to T Graphs ....................................................................................... 06

References & Bibliography ........................................................................................... 07


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Definitions Related to T Graphs

o Cycle GraphDefinition: The cycle or circle graph Cnis the graph with n vertices where each vertex is

connected to exactly two edges. [1]

Eg: C4

o Complete GraphDefinition: The complete graph Knis the graph with n vertices where each pair of graph

vertices is connected with an edge. [2]

Eg: K4

o Triangle GraphDefinition: The triangle graph is thecycle graph C3, which is also the complete graph K3. [3]

o Base T GraphDefinition: The base T graph can be defined as equivalent to the triangle graph.
http://mathworld.wolfram.com/CycleGraph.htmlhttp://mathworld.wolfram.com/CycleGraph.html


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o Border of a T GraphDefinition: If a vertex of a T Graph has only two edges connected, then that vertex is said to

be on the border of the T Graph.

Eg:

o n-Coloured GraphDefinition: A graph can be n-coloured if there is a way to colour all the vertices of the graph

using only n colours, such that two vertices sharing a common edge do not have the same

colour.

(Definition modified from [4])

Vertices of Border


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Problem 09T Graphs

Define a T Graph recursively as follows:

This is a T Graph

If G is a T Graph and v is a vertex of G, then this is also a T Graph.

a) Draw an example of a T Graph with 7 vertices.Solution:

vG


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b) Prove that every vertex in a T Graph has even degree.Solution:

Statement: Every vertex in a T Graph has even degree.

Proof by Induction (on the number of base triangles n, in the T graph)

Base Case: Let n = 1. Then the T Graph reduces to a triangle graph, where each of thethree vertices is connected with the two other vertices. So trivially every vertex has

even degree.

Inductive Hypothesis (IH): Assume every vertex in any T graph with n = k-1(for k >1) base triangles has even degree.

Inductive Step: Prove true for n = k. Let G be a T graph with n = k-1 base triangles. As mentioned in the IH,

every vertex in G has even degree.

According to the definition of T Graphs, the T Graph with n = k basetriangles is derived from G by adding one more base triangle where one

vertex is common to both G and the newly added triangle.( Let the

common vertex be v and the newly added triangle, M )

From the base case, every vertex in M has even degree.( degree = 2 ) When considering v, the common vertex, before connecting M it had even

degree. After connecting M, two more edges will be connected to v. So the

degree of v will be increased by two. An even number added to an even

number results in an even number. Hence every vertex in the resulting T

Graph (with n = k base triangles) also has even degree.

vG

M


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c) Prove that any T Graph can be three-coloured.

Statement: Any T Graph can be three-coloured.

Proof:

A T Graph can be described as a collection of interconnected Base T Graphs whichhave only one vertex common between any two of them.

If we consider any vertex in a T Graph, it can be a vertex common to several Base TGraphs or a vertex belonging to a single Base T Graph.

If we colour the common vertex with a particular colour, it is necessary to have twodistinct colours to colour each vertex pair of each separate Base T Graph connected to

the common vertex.

This colouring procedure can be started with any vertex in the graph and shouldcontinue until we meet all the borders of the T graph.

Starting

Vertex

1 2

3


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Problems of T Graphs

Application Problem related to T Graphs:

A girl bought 5 dark chocolates from a candy shop. The owner of that shop told her, that if she

returns 2 chocolate covers of dark chocolate or return a dark chocolate cover while showing a milk

chocolate cover, he will give a new milk chocolate. But to get a new milk chocolate, she cant show

2 milk chocolate covers at the same time.

What is the maximumnumber of milk chocolate covers that will be in her hand?

True/False questions related to T Graphs:

1. Given the definition of a triangle graph as mentioned in the definitions sect ion [see page1], aT Graph can be created by connecting several base triangle graphs together.

2. Definition: If u and v are two vertices in a graph G, a u-v walk is an alternating sequence ofvertices and edges starting with u and ending at v. Consecutive vertices and edges are

incident.

Definition: The number of edges in a walk gives its length.

Definition: A path in a graph is a walk with no repeated vertices.

Definition: A cycle in a graph is defined as a path of length >= 3, whose start and end

vertices are the same. [1]

Given the definition of a cycle as above, only cycles of length three can be found in a T

graph.


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References

[1] Rapti de Silva (2013, June 18). Course Overview and Introduction[Online]. Available:

http://lms.uom.lk/moodle192/mod/resource/view.php?id=25832

[2] Wolfram Math World (2013, July 2). Complete Graph [Online]. Available:http://mathworld.wolfram.com/CompleteGraph.html

[3] Wolfram Math World (2013, July 2). Triangle Graph [Online]. Available:

http://mathworld.wolfram.com/TriangleGraph.html

[4] Rapti de Silva (2013, June 25).Mathematical Thinking & Big Ideas in Graph Theory[Online].

Available:http://lms.uom.lk/moodle192/mod/resource/view.php?id=26066

Additional Bibliography

Wolfram Math World (2013, July 16).K-Colourable Graph [Online]. Available:http://mathworld.wolfram.com/k-ColorableGraph.html
http://lms.uom.lk/moodle192/mod/resource/view.php?id=25832http://lms.uom.lk/moodle192/mod/resource/view.php?id=25832http://mathworld.wolfram.com/CompleteGraph.htmlhttp://mathworld.wolfram.com/CompleteGraph.htmlhttp://mathworld.wolfram.com/TriangleGraph.htmlhttp://mathworld.wolfram.com/TriangleGraph.htmlhttp://lms.uom.lk/moodle192/mod/resource/view.php?id=26066http://lms.uom.lk/moodle192/mod/resource/view.php?id=26066http://lms.uom.lk/moodle192/mod/resource/view.php?id=26066http://mathworld.wolfram.com/k-ColorableGraph.htmlhttp://mathworld.wolfram.com/k-ColorableGraph.htmlhttp://mathworld.wolfram.com/k-ColorableGraph.htmlhttp://lms.uom.lk/moodle192/mod/resource/view.php?id=26066http://mathworld.wolfram.com/TriangleGraph.htmlhttp://mathworld.wolfram.com/CompleteGraph.htmlhttp://lms.uom.lk/moodle192/mod/resource/view.php?id=25832

Group 9 Research Report 1

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