Euler Path Calculator
Determine whether an Euler path or circuit exists using vertex degree conditions.
Inputs
Format: X-Y separated by semicolons
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
Values used
Vertices (comma separated) = A, B, C, D; Edges (semicolon separated pairs) = A-B; A-C; A-D; B-C; C-D
Euler Circuit
Exists iff the graph is connected and every vertex has even degree
Euler Path
Exists iff the graph is connected and has exactly 0 or 2 vertices of odd degree
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
How it works
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
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.