Euler Circuits & the Even-Degree Test | Discrete Mathematics §10.5

In 1736 Euler settled a puzzle the townspeople of Königsberg had argued over for years: could you cross all seven bridges exactly once and return home? His answer launched graph theory — and it comes down to counting odd vertices. In this video: Euler circuits, on the board. An Euler circuit uses every edge exactly once and returns to its start. We trace one on a bowtie graph in a single stroke (revisiting a vertex is fine — it's edges we can't repeat), then derive the even-degree test: each pass through a vertex uses one edge in and one out, so every vertex must have even degree. Finally we apply it to Königsberg — four odd-degree vertices, so no such route exists. Then continue with §10.5 Euler Paths and Building One (next in the playlist). This video is part of Discrete Mathematics · Graphs (§10.5 — Euler & Hamilton Paths). Full section playlist linked above / in the description on the channel. Made with the Engineering Simplified method: a calm, two-voice story lesson taught on a chalk-and-board, with every definition and example drawn out step by step. Topic coverage follows Rosen, Discrete Mathematics and Its Applications (Chapter 10).