📐 Eulerian Circuit Characterization

A connected graph $G$ has an Eulerian circuit if and only if every vertex of $G$ has even degree.

Proof: Necessity: Each visit to a vertex uses two edges (one in, one out), so degrees must be even. Sufficiency: Start at any vertex and walk without repeating edges. Since degrees are even, you can only get stuck at the start. If edges remain, find a vertex on the circuit adjacent to an unused edge, bu...

From: Introduction to Graph Theory

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