Euler Path Calculator
Determine whether an Euler path or circuit exists using vertex degree conditions.
About Euler Path Calculator
An Euler path traverses every edge of a graph exactly once. An Euler circuit is an Euler path that starts and ends at the same vertex. The conditions are simple: a connected graph has an Euler circuit if all vertices have even degree, and an Euler path if exactly 0 or 2 vertices have odd degree.
Formulas
Euler Circuit
Exists iff the graph is connected and every vertex has even degree
- degree
- Number of edges incident to a vertex
Euler Path
Exists iff the graph is connected and has exactly 0 or 2 vertices of odd degree
Worked example
With the default values below, Euler Path Calculator returns Euler Path Exists? of Yes (start at A, end at C). Change any input above and every figure here is recalculated in your browser.
Inputs used
- Vertices (comma separated)
- A, B, C, D
- Edges (semicolon separated pairs)
- A-B; A-C; A-D; B-C; C-D
Results
- Euler Path Exists?
- Yes (start at A, end at C)
- Euler Circuit Exists?
- No (two vertices have odd degree)
- Odd-degree Vertices
- A, C
- Vertex Degrees
- A:3, B:2, C:3, D:2
Step by step
- 1Values used
- 2Euler Circuit
- 3Euler Path
- 4Euler Path Exists?= Yes (start at A, end at C)
- 5Euler Circuit Exists?= No (two vertices have odd degree)
- 6Odd-degree Vertices= A, C
- 7Vertex Degrees= A:3, B:2, C:3, D:2
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the difference between Euler and Hamiltonian paths?
An Euler path traverses every edge exactly once (vertices may repeat). A Hamiltonian path visits every vertex exactly once (edges may be skipped).
Why does the Königsberg bridge problem have no solution?
The graph of Königsberg's bridges has four vertices, all with odd degree. An Euler path requires 0 or 2 odd-degree vertices, so no solution exists.