Determine whether an Euler path or circuit exists using vertex degree conditions.
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.
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
An Euler path traverses every edge exactly once (vertices may repeat). A Hamiltonian path visits every vertex exactly once (edges may be skipped).
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.