CC222806 Research Methods in Geometry II

Hiroshima University Syllabus

Japanese

Academic Year 2024Year School/Graduate School School of Education

Lecture Code CC222806 Subject Classification Specialized Education

Subject Name 幾何学研究法II

Subject Name
（Katakana）キカガクケンキュウホウII

Subject Name in
English Research Methods in Geometry II

Instructor TERAGAITO MASAKAZU

Instructor
(Katakana) テラガイト　マサカズ

Campus Higashi-Hiroshima Semester/Term 3rd-Year,  Second Semester,  4Term

Days, Periods, and Classrooms (4T) Tues5-8：EDU L107

Lesson Style Lecture Lesson Style
(More Details) 　

Credits 2.0 Class Hours/Week 　 Language of Instruction J : Japanese

Course Level 3 : Undergraduate High-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Students of Faculty of Education , Especially those of the Program in Mathematics Education

Keywords Graph theory, discrete mathematics

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students)
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Secondary School Mathematics Education
（Knowledge and Understanding）
・To understand basic knowledge of teaching contents of mathematic education.
（Abilities and Skills）
・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.

Class Objectives
/Class Outline Objectives and Theme
・Know major topic of theories of graph.
・Know examples of applying graphs to other fields, day-to day life, and society  life.

Course Overview
This course will deal with the theory of graph which is illustrated as an example of ‘ utilization of mathematics’ in high school education, as a main topic, including problems of single-stroke drawing, Hamilton cycle, color-coding of a map.
One of the good things about the graph theory is that it requires almost no background knowledge. A graph is merely an assembly of discrete data. However, it can be seen as a one-dimensional diagram. Another advantage is that a graph can be seen as a concrete and visual object in the modern mathematics which is quite abstract.

Class Schedule Lesson1: Graph, Handshaking Lemma, Degree Sequence

Lesson2: Subgraph, Isomorphism

Lesson3: Distance on graphs

Lesson4: Trees

Lesson5: Hamiltonian graph

Lesson6: Ramsey theory

Lesson7: Planar Graph

Lesson8: Vertex coloring

Lesson9: Four-Color Theorem

Lesson10: Achromatic number, chromatic polynomial

Lesson11: Crossing number and genus

Lesson12: Labelling
Lesson13: Algebraic graph theory (Adjacency matrix, spectrum)

Lesson14: Algebraic graph theory (Laplacian)

Lesson15: Other topics in graph theory

There will be a final exam in Lesson 16.

Text/Reference
Books,etc. There will be no textbooks for this course. However there are a number of Japanese books about graph theory. You should borrow some from library.
There are also a number of good English books. The following shows the recommend English books:

Pearls in Graph Theory，N.Hartsfield and G.Ringel，Academic Press
Graph Theory, J.A.Bondy and U.S.Murty, Springer
Graph Theory, R.Diestel, Springer

PC or AV used in
Class,etc. 　

(More Details)

Learning techniques to be incorporated 　

Suggestions on
Preparation and
Review Lesson 1：Graph, Handshaking Lemma, Degree Sequence
What is a graph? After explaining its definition, the instructor will introduce handshaking lemma, which is easy but basic and important. The lesson will also touch upon degree sequence.

Lesson 2: Subgraph, Isomorphism
Following Lesson1, this lesson will continue to introduce concepts which serve as the bases of theories.

Lesson 3: Distance on graphs
We introduce a distance on a graph.

Lesson 4: Tree
Among different kinds of graphs, ‘ tree’ is the most basic graph. The lesson will explain distinctive properties of trees.

Lesson 5: Hamiltonian graph
There is a concept called Hamiltonian graph which is a problem of whether it is possible to visit every vertex once and come back to the starting vertex. Characterization of Hamiltonian graph is not yet resolved. This is an important problem in the graph theory.

Lesson 6: Ramsey Theory
We deal with a theme called Ramsey theory.
This can be utilized a good teaching material.

Lesson 7: Planar Graph
When we deal with a graph as an object of geometry, complexity of the graph can be measured with a criterion of whether it can be embedded in a plane or not.

Lesson 8: Vertex Coloring
Coloring Problems are a big theme of the graph theory.

Lesson 9: Four-color theorem
Four-color problem which is a well-known theorem about color-coding of a map falls into this category.  We will study an assignment of colors to vertices and an assignment of colors to edges.

Lesson 10: Achromatic number and chromatic polynomial
Related to vertex coloring, we introduce achromatic number and chromatic polynomial.

Lesson 11: Crossing number and genus
A graph that cannot be embedded in a plane surface has a problem of how many intersections of edges can be reduced, which generates the concept of crossing number.
A graph that cannot be embedded in a plane will be embedded if you change from a plane or sphere to more general surface. This is a theme of a field called topological graph theory.

Lesson 12: Labeling
There is a theme called labeling of a graph. This is problem like puzzle. Researcher who engages in this is called ``labeler".
The lessons will introduce magic labeling that is related to magic square, and graceful labeling which is beautiful labeling, and others.

Lesson 13-14: Algebraic graph theory
For a graph, we can assign several matrices. Using their spectrum, we examine graphs.
This is the main theme of algebraic graph theory.

Lesson 15: Other topics
Graph theory consists of a lot of topics. In the last lesson, we treat several topics.

Requirements The good thing about this course is that this course does not require background knowledge.
Most of proof questions are by induction and contradiction.
Students should take this course considering applying it to high school mathematics.

Grading Method Students will be evaluated based on final exam (100% ).

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message Needless to say, students in the 3rd year of the Faculty of Education should not choose to sit in the back row.
Since you have already experienced teaching practice and know the standpoint of a teacher, you should attend the class thinking about the standpoint of a teacher and how you can contribute to the class as a student. That should be a difference from students of other faculties.

Other

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2024Year	School/Graduate School	School of Education
Lecture Code	CC222806	Subject Classification	Specialized Education
Subject Name	幾何学研究法II
Subject Name （Katakana）	キカガクケンキュウホウII
Subject Name in English	Research Methods in Geometry II
Instructor	TERAGAITO MASAKAZU
Instructor (Katakana)	テラガイト　マサカズ
Campus	Higashi-Hiroshima	Semester/Term	3rd-Year, Second Semester, 4Term
Days, Periods, and Classrooms	(4T) Tues5-8：EDU L107
Lesson Style	Lecture	Lesson Style (More Details)

Credits	2.0	Class Hours/Week		Language of Instruction	J : Japanese
Course Level	3 : Undergraduate High-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Students of Faculty of Education , Especially those of the Program in Mathematics Education
Keywords	Graph theory, discrete mathematics
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Secondary School Mathematics Education （Knowledge and Understanding）・To understand basic knowledge of teaching contents of mathematic education. （Abilities and Skills）・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.
Class Objectives /Class Outline	Objectives and Theme ・Know major topic of theories of graph. ・Know examples of applying graphs to other fields, day-to day life, and society life. Course Overview This course will deal with the theory of graph which is illustrated as an example of ‘ utilization of mathematics’ in high school education, as a main topic, including problems of single-stroke drawing, Hamilton cycle, color-coding of a map. One of the good things about the graph theory is that it requires almost no background knowledge. A graph is merely an assembly of discrete data. However, it can be seen as a one-dimensional diagram. Another advantage is that a graph can be seen as a concrete and visual object in the modern mathematics which is quite abstract.
Class Schedule	Lesson1: Graph, Handshaking Lemma, Degree Sequence Lesson2: Subgraph, Isomorphism Lesson3: Distance on graphs Lesson4: Trees Lesson5: Hamiltonian graph Lesson6: Ramsey theory Lesson7: Planar Graph Lesson8: Vertex coloring Lesson9: Four-Color Theorem Lesson10: Achromatic number, chromatic polynomial Lesson11: Crossing number and genus Lesson12: Labelling Lesson13: Algebraic graph theory (Adjacency matrix, spectrum) Lesson14: Algebraic graph theory (Laplacian) Lesson15: Other topics in graph theory There will be a final exam in Lesson 16.
Text/Reference Books,etc.	There will be no textbooks for this course. However there are a number of Japanese books about graph theory. You should borrow some from library. There are also a number of good English books. The following shows the recommend English books: Pearls in Graph Theory，N.Hartsfield and G.Ringel，Academic Press Graph Theory, J.A.Bondy and U.S.Murty, Springer Graph Theory, R.Diestel, Springer
PC or AV used in Class,etc.
(More Details)
Learning techniques to be incorporated
Suggestions on Preparation and Review	Lesson 1：Graph, Handshaking Lemma, Degree Sequence What is a graph? After explaining its definition, the instructor will introduce handshaking lemma, which is easy but basic and important. The lesson will also touch upon degree sequence. Lesson 2: Subgraph, Isomorphism Following Lesson1, this lesson will continue to introduce concepts which serve as the bases of theories. Lesson 3: Distance on graphs We introduce a distance on a graph. Lesson 4: Tree Among different kinds of graphs, ‘ tree’ is the most basic graph. The lesson will explain distinctive properties of trees. Lesson 5: Hamiltonian graph There is a concept called Hamiltonian graph which is a problem of whether it is possible to visit every vertex once and come back to the starting vertex. Characterization of Hamiltonian graph is not yet resolved. This is an important problem in the graph theory. Lesson 6: Ramsey Theory We deal with a theme called Ramsey theory. This can be utilized a good teaching material. Lesson 7: Planar Graph When we deal with a graph as an object of geometry, complexity of the graph can be measured with a criterion of whether it can be embedded in a plane or not. Lesson 8: Vertex Coloring Coloring Problems are a big theme of the graph theory. Lesson 9: Four-color theorem Four-color problem which is a well-known theorem about color-coding of a map falls into this category. We will study an assignment of colors to vertices and an assignment of colors to edges. Lesson 10: Achromatic number and chromatic polynomial Related to vertex coloring, we introduce achromatic number and chromatic polynomial. Lesson 11: Crossing number and genus A graph that cannot be embedded in a plane surface has a problem of how many intersections of edges can be reduced, which generates the concept of crossing number. A graph that cannot be embedded in a plane will be embedded if you change from a plane or sphere to more general surface. This is a theme of a field called topological graph theory. Lesson 12: Labeling There is a theme called labeling of a graph. This is problem like puzzle. Researcher who engages in this is called ``labeler". The lessons will introduce magic labeling that is related to magic square, and graceful labeling which is beautiful labeling, and others. Lesson 13-14: Algebraic graph theory For a graph, we can assign several matrices. Using their spectrum, we examine graphs. This is the main theme of algebraic graph theory. Lesson 15: Other topics Graph theory consists of a lot of topics. In the last lesson, we treat several topics.
Requirements	The good thing about this course is that this course does not require background knowledge. Most of proof questions are by induction and contradiction. Students should take this course considering applying it to high school mathematics.
Grading Method	Students will be evaluated based on final exam (100% ).
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message	Needless to say, students in the 3rd year of the Faculty of Education should not choose to sit in the back row. Since you have already experienced teaching practice and know the standpoint of a teacher, you should attend the class thinking about the standpoint of a teacher and how you can contribute to the class as a student. That should be a difference from students of other faculties.
Other
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

Back to syllabus main page