MATH1007: Graph Theory Bipartite Graphs, Eulerian Paths & Degree Sequences

Skills Exercise

Questions

1.Explain why the following graph is not bipartite. Show that, by removing a single edge, we can make it bipartite.

1. Explain why the graph above does not have an Eulerian path. Show that, by removing a single edge, we can get a graph which does have an Eulerian path.

2. Do there exist trees with the following degree sequences? If you answer “yes”, you should explicitly write down a tree. If you answer “no”, you should justify why it is impossible. (a) 4, 4, 2, 2, 1 (b) 2, 2, 2, 2, 0 (c) 3, 2, 1, 1, 1 (d) 2, 1, 1, 1, 1

3. A connected, simple graph ???? has degree sequence 5, 5, 5, 4, 4, 3.

1. How many edges does ???? have? Justify your answer.
2. If ???? had a planar drawing, how many faces would it have? Justify your answer.
3. If ???? had a planar drawing, what would be

4. Consider the following syntax graph:

1. Write down the bracketed arithmetic expression which this tree describes.
2. By taking a suitable traversal of the syntax tree, write down this arithmetic expression in reverse Polish notation. Explain how you got your traversal.
3. Use a stack to evaluate the reverse Polish expression from (b). Is this the answer that you expect from (a)

5. Complete the following run of Dijkstra’s algorithm rooted at the vertex ????. At each step, you should indicate the vertex you have selected to add to the spanning tree, as well as the vertices you have removed from the fringe. Illustrate your final spanning tree on a copy of the graph.

Module 3

1. (a) Run Euclid’s algorithm to determine the greatest common divisor ???? of 602 and 217. (b) Now run the extended Euclidean algorithm in order to find integers ???? and ???? such that

???? × 602 + ???? × 217 = ???? .

2. A class of 30 students has 15 students from Senegal, 10 students from Tanzania and 5 students from Uganda. A committee of 6 students is required.

1. How many different committees are possible?
2. If the committee needs 3 students from Senegal, 2 students from Tanzania, and 1 from Uganda how many different committees are possible?
3. How many different committees are possible if the committee has at least 5 representatives from Senegal?

Summary of Assessment Requirements

This assessment required students to demonstrate their understanding of key Graph Theory and Discrete Mathematics concepts through a set of structured questions. The tasks were designed to assess the ability to apply theoretical knowledge to problem-solving scenarios using logical reasoning and mathematical justification.

The key components covered in the assessment included:

• Graph Theory Applications Understanding bipartite graphs, Eulerian paths, and degree sequences.
• Tree Structures and Degree Validation Determining the feasibility of trees with given degree sequences.
• Planar Graphs and Euler’s Formula Computing the number of edges and faces in connected planar graphs.
• Syntax Trees and Arithmetic Expressions Translating between bracketed expressions, syntax trees, and reverse Polish notation.
• Shortest Path Algorithms Executing Dijkstra’s algorithm to construct a minimum spanning tree.
• Euclid’s and Extended Euclidean Algorithm Finding greatest common divisors and integer coefficients.
• Combinatorics Applications Calculating possible combinations under different selection constraints.

The assessment aimed to evaluate not only computational skills but also the student’s ability to explain mathematical reasoning clearly, justify each answer, and apply algorithms accurately.

Mentor’s Step-by-Step Guidance and Approach

The academic mentor guided the student through the assessment by structuring the learning and problem-solving process in logical, progressive steps to reinforce conceptual clarity and analytical thinking.

Step 1: Understanding Bipartite and Eulerian Graphs

The mentor began by revising the definitions and properties of bipartite and Eulerian graphs. The student was encouraged to visualize the given graph, identify odd-length cycles (which violate bipartite conditions), and explore how removing one edge could resolve the issue. For the Eulerian path, the mentor explained how to identify vertices with odd degrees and demonstrated that removing one edge can result in exactly two vertices of odd degree, satisfying the Eulerian path condition.

Step 2: Degree Sequences and Tree Feasibility

In this part, the mentor revisited the handshaking lemma and the property that the sum of degrees in a tree equals 2(n1). The student practiced verifying each given degree sequence and constructing valid trees when possible, thereby strengthening reasoning skills in graph construction and logical proof.

Step 3: Planar Graphs and Euler’s Formula

The mentor explained how to determine the number of edges using the formula Sum of degrees=2E\text{Sum of degrees} = 2ESum of degrees=2E and then applied Euler’s formula (V E + F = 2) to find the number of faces. The mentor emphasized conceptual understanding of planar graph constraints and how degree distribution affects planarity.

Step 4: Syntax Trees and Reverse Polish Notation (RPN)

To build algorithmic reasoning, the mentor illustrated how inorder, preorder, and postorder traversals correspond to different expression formats. The student converted a syntax tree to bracketed expression and then derived its RPN using postorder traversal, followed by stack-based evaluation for verification. This step linked theoretical graph structures with computational implementation logic.

Step 5: Dijkstra’s Algorithm Execution

The mentor guided the student through each iteration of Dijkstra’s algorithm, identifying the vertex with minimum tentative distance, updating neighbors, and constructing the shortest-path tree. The student was encouraged to represent each step in a table for clarity and accuracy.

Step 6: Euclid’s and Extended Euclidean Algorithm

The mentor reviewed both the standard Euclidean method for finding GCD and its extended form for obtaining coefficients (s, t) satisfying s×a+t×b=gcd(a,b)s \times a + t \ × b = \text{gcd}(a,b)s×a+t×b=gcd(a,b). This exercise reinforced procedural accuracy and algebraic understanding.

Step 7: Combinatorics and Probability Applications

In the final section, the mentor introduced combinations (nCr) to calculate possible committee arrangements under different constraints. Each sub-question was used to strengthen understanding of counting principles and conditional combinations.

Outcome and Learning Objectives Achieved

By the end of the assessment, the student had:

• Strengthened analytical reasoning and problem-solving abilities across graph theory, algorithms, and combinatorics.
• Demonstrated proficiency in applying mathematical logic and computational steps accurately.
• Developed the ability to explain theoretical concepts and justify each answer clearly.
• Gained practical understanding of algorithmic thinking (Dijkstra’s, Euclid’s) and data representation (syntax trees, RPN).
• Met core learning objectives of critical analysis, mathematical application, and theoretical comprehension within discrete mathematics.

Get Expert Guidance for Your Assignment Success

Looking for a clear example of how to approach this task? You can download the sample solution below to understand the correct structure, formatting, and academic style required. This sample is designed strictly for reference and learning purposes only submitting it as your own work may lead to plagiarism issues or academic penalties.

If you want a 100% original, plagiarism-free, and well-researched assignment, our professional academic writers can prepare a custom solution tailored to your topic and university guidelines. Every order comes with proper citations, clear explanations, and guaranteed confidentiality helping you achieve top grades with confidence.

Why Choose a Fresh Custom Solution?

• Written from scratch by qualified academic experts
• Completely plagiarism-free and Turnitin-safe
• Customized to your specific requirements and marking criteria
• Delivered on time with full editing and formatting support

Plagiarism Disclaimer:
The provided sample is for educational reference only. Do not submit it as your own academic work. Always use it to understand structure, content flow, and analytical approach.

Take the next step towards academic excellence:
Download Sample Solution  Order Fresh Assignment