Eulerian and Hamiltonian Paths
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# Eulerian and Hamiltonian Paths ## 7.1 Eulerian Trails An **Eulerian trail** uses every edge exactly once. An **Eulerian circuit** starts and ends at the same vertex.
Eulerian and Hamiltonian Paths
7.1 Eulerian Trails
An Eulerian trail uses every edge exactly once. An Eulerian circuit starts and ends at the same vertex.
Euler's Theorem
- A connected graph has an Eulerian circuit iff all vertices have even degree.
- A connected graph has an Eulerian trail (not circuit) iff exactly two vertices have odd degree.
7.2 Hamiltonian Paths
A Hamiltonian path visits every vertex exactly once. A Hamiltonian cycle starts and ends at the same vertex.
Dirac's Theorem
If is a graph with vertices and every vertex has degree , then has a Hamiltonian cycle.
Ore's Theorem
If is a graph with and for every non-adjacent pair , then has a Hamiltonian cycle.
✅ Practice Questions
Q1: Does have an Eulerian circuit? A Hamiltonian cycle?
Solution: every vertex has degree 4 (even), so Eulerian circuit exists. Since and , by Dirac, Hamiltonian cycle exists. Q2: The Königsberg bridge problem — 7 bridges connecting 4 land masses. Does an Eulerian trail exist? SolutionThe graph had 4 vertices with degrees 3,3,3,3 (all odd). Exactly 4 odd-degree vertices (not 0 or 2), so no Eulerian trail exists. Join Discord PreviousGraph Coloring AppsNextGraph Algorithms