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Traversable (Euler-Trail) Graphs

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Traversable Graphs

Study Guide

Key Definition

A connected graph has an Euler trail (is traversable) if and only if it has exactly 0 or 2 vertices of odd degree.

Important Notes

A graph is traversable if you can draw it without lifting the pencil and without retracing any edge (has an Euler trail).
The graph must be connected (ignoring isolated vertices).
Each vertex must have even degree, except for at most two vertices of odd degree.
Graphs with more than two odd-degree vertices cannot have an Euler trail.
Understanding the degree of vertices is key to determining Euler trails.

Mathematical Notation

deg(v) denotes the number of edges incident to vertex v

Remember to use proper notation when solving problems

Why It Works

By Euler's theorem, a connected graph has an Euler trail precisely when it has zero or two vertices of odd degree.

Remember

Connectivity and the count of odd-degree vertices (0 or 2) determine the existence of an Euler trail.

Quick Reference

Euler's Theorem:Connected graph with 0 or 2 odd-degree vertices has an Euler trail

Understanding Euler Trails in Graphs

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An Euler trail is a path that uses every edge exactly once. For a connected graph, it exists if exactly 0 or 2 vertices have odd degree. If 0, it is a circuit; if 2, the trail starts and ends at the odd vertices.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In a connected graph with vertex degrees 2, 3, 3, 4, does it have an Euler trail?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Four towns A, B, C, D are connected by roads forming a connected graph. The degrees are deg(A)=3, deg(B)=2, deg(C)=1, deg(D)=2. Determine if an Euler trail exists.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Can a connected graph with degrees 1, 3, 3, 2, 4 have an Euler trail?

Challenge Quiz

Single Choice Quiz

Advanced

A connected graph has vertices with degrees 3, 3, 3, 3, 2, 2. Is there an Euler trail?

Recap

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Review key concepts and takeaways