Post on 13-Apr-2018
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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.html7/27/2019 Group 9 Research Report 1
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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