Question

Which trees have an Eulerian path?

   Which trees have an Eulerian path?

Introductory Combinatorics

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 11, Problem 54

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Solved by Expert Victoria Dollar

08/04/2020

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A graph has an Eulerian path if and only if it is connected and has exactly 0 or 2 vertices with an odd degree (number of edges connected to the vertex). Show more…

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Which trees have an Eulerian path?

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Key Concepts

Eulerian Path

An Eulerian path in a graph is a trail which visits every edge exactly once. For such a path to exist, the graph must be connected, and it can have at most two vertices of odd degree. This concept is central in graph theory and is used to determine whether a graph can be traversed in a single continuous path without retracing any edge.

Tree in Graph Theory

A tree is a connected graph with no cycles, characterized by having exactly one less edge than its number of vertices. Trees have a hierarchical structure and are fundamental in graph theory, providing a simplified model where many properties, such as vertex degrees and connectivity, can be studied in a clear and intuitive way.

Degree Conditions for Eulerian Paths

A key criterion for the existence of an Eulerian path in any graph is based on the degrees of its vertices: either no vertices or exactly two vertices can have an odd degree. In the context of trees, because trees typically have at least two leaves (vertices of degree one), only those trees that have exactly two vertices of odd degree – essentially, linear path structures – can possess an Eulerian path.

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