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Understanding Euler and Hamilton Paths in Undirected Graphs

This module delves into the concepts of Euler and Hamilton paths within undirected graphs. An Euler path traverses every edge exactly once, while an Euler circuit starts and finishes at the same vertex. Identifying Euler paths requires analyzing the degree of vertices—Euler paths exist with exactly two odd-degree vertices. On the other hand, Hamilton paths touch each vertex exactly once without needing to cover all edges. Unlike Euler paths, there are no strict conditions for Hamilton paths, making their identification more about exploration.

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Understanding Euler and Hamilton Paths in Undirected Graphs

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  1. Module 5 – Networks and Decision Mathematics Chapter 23 – Undirected Graphs

  2. 23.4 Euler and Hamilton Paths • A path can be considered as a sequence of edges of the form AB, BG, GE, EC, CD, DE, EG, GF for the graph below: • Insert graph • A circuit is a sequence of edges linking successive vertices that starts and ends at the same vertex. For the graph below: • Insert graph • A path that includes every edge just once is called an Euler Path. • An Euler Circuit is an Euler Path that starts and ends at the same vertex.

  3. Identifying Euler Paths & Circuits • To identify Euler Circuits, look for connected graphs where all vertices have an even degree. • An Euler Path exists if there are exactly 2 odd degree vertices. The path would start at one of these and end at the other. • Insert example • For an Euler path, you would start at B and end at C or vice versa. ie B-A-E-D-B-C-D-C • Note: With Euler paths/circuits, edges cannot be travelled over more than once but you can revisit a vertex.

  4. Identifying Hamilton Paths & Circuits • A Hamilton Path is a path through a graph that passes through each vertex exactly once. • Insert example • A Hamilton Path could be: A-F-G-E-B-C-D-H (there are many more) • A Hamilton Circuit is a Hamilton Path that starts and ends at the same vertex. For the previous graph, a Hamilton Circuit could be: C-D-H-G-F-A-E-B-C • Note: Not all edges must be covered but all vertices must be visited.

  5. Remember:Euler Paths/Circuits focus on EDGES and Hamilton Paths/Circuits focus on VERTICES. • Unlike Euler, there are no specific conditions (ie 2 odd vertices) that are used to identify Hamilton paths or circuits. It is simply a matter of trial and error.

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