Eulerian and Hamiltonian Walks (HSC SSCE Mathematics Standard): Flashcards

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Eulerian and Hamiltonian walks
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Eulerian walks - which graph element?

Edges (every edge exactly once)

Hamiltonian walks - which graph element?

Vertices (every vertex exactly once)

Eulerian trail definition

Uses every edge once, different start/end vertices

Eulerian circuit definition

Uses every edge once, same start/end vertex

Eulerian trail existence condition

Connected graph with exactly 2 odd-degree vertices

Eulerian circuit existence condition

Connected graph where all vertices have even degree

Hamiltonian path definition

Visits every vertex once and only once

Hamiltonian cycle definition

Hamiltonian path that starts & ends at same vertex

How to identify Eulerian walks

Use vertex degree rules (count odd-degree vertices)

How to identify Hamiltonian walks

No simple rule - inspect graph carefully

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