You voted, and the winner is Chapter 10: Graphs! This is one of the most applicable and visual sections of Discrete Mathematics. Whether we are modeling the NFL season schedules, optimizing a mail carrier's route, or coloring a map, graph theory provides the framework we need.

What is a Graph?

At its core, a graph $G = (V, E)$ consists of a set of vertices $V$ (nodes) and a set of edges $E$ that connect them. Graphs can be:

  • Undirected: The relationship is symmetric (e.g., a handshake).
  • Directed (Digraphs): The relationship has a direction (e.g., a one-way street or a Twitter follow).

Key Concepts in Chapter 10

The attached notes and examples cover several critical topics. Here is a breakdown of what you need to focus on:

1. Graph Isomorphism (Section 10.3)

How do we tell if two graphs are actually the "same" structure, just drawn differently? We look for an isomorphism. To be isomorphic, two graphs must have the same graph invariants, such as the same number of vertices, edges, and degree sequences.

From the Extra Examples: Just because two graphs have the same degree sequence doesn't automatically mean they are isomorphic; you often need to check the adjacency of specific vertices.

2. Euler vs. Hamilton Paths (Section 10.5)

This section often trips students up, so remember this distinction:

  • Euler Circuit/Path: Focuses on edges. You want to cross every bridge (edge) exactly once. A connected graph has an Euler circuit if and only if every vertex has an even degree.
  • Hamilton Circuit/Path: Focuses on vertices. You want to visit every city (vertex) exactly once. There is no simple "if and only if" rule for this, making it a harder problem to solve computationally.

Application: The Chinese Postman Problem (discussed in the notes) asks for the shortest tour that covers every street—an optimization of the Euler path concept involving "deadheading" (retracing steps).

3. Planar Graphs & Euler's Formula (Section 10.7)

A graph is planar if it can be drawn without any edges crossing. For any connected planar simple graph, Euler's Formula holds true:

$$v - e + r = 2$$

Where $v$ is the number of vertices, $e$ is the number of edges, and $r$ is the number of regions (including the infinite unbounded region).

4. Graph Coloring (Section 10.8)

This deals with assigning values (colors) to elements of a graph subject to certain constraints. The chromatic number, denoted $\chi(G)$, is the least number of colors needed to color the vertices so that no two adjacent vertices share the same color. By the Four Color Theorem, any planar map requires no more than 4 colors.

Study Resources

I have attached the full solutions and extra examples below. Please pay special attention to the adjacency matrices examples, as they are a standard way to represent graphs for computer algorithms.

Files Included:

  • Chapter 10 Notes & Solutions
  • Extra Examples (Graph Models, Isomorphism, Euler/Hamilton Paths, Planarity, Coloring)
  • Answers for Even Problems

Keep practicing those proofs, and happy graphing!