Planar Graphs: Drawing Without Crossings | Discrete Mathematics §10.7

A classic puzzle asks you to connect three houses to three utilities without any pipe crossing another. Generations have tried and failed — and the reason why turns out to be one of the deepest facts in graph theory. In this video: planar graphs, on the board. A graph is planar if it can be drawn in the plane with no edges crossing. We redraw the complete graph on four vertices to remove its crossing (a triangle with a center), and untangle the cube by nesting one square inside another. We count the regions such a drawing carves the plane into — including the unbounded outer region. Then we tackle the three-houses-three-utilities graph and argue, region by region, that the last connection is always trapped — so it is nonplanar. Then continue with §10.7 Euler's Formula and the Planar Inequalities (next in the playlist). This video is part of Discrete Mathematics · Graphs (§10.7 — Planar Graphs). 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).