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Eulerian paths — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Königsberg as a graph. The four landmasses as circles and the seven bridges as edges; every circle has an odd number of edges.

    Seven bridges, and the invention of throwing things away

    Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.

    part 1 · discrete
  2. Hierholzer's construction on 6 vertices and 9 edges. Three views of one graph whose vertices all have even degree: a first closed walk that stops back at its start, the loops walked from vertices on it with edges left over, and the single circuit made by splicing them, with every edge numbered in order.

    A walk that splices in its own detours

    Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.

    part 2 · discrete
  3. The postman's route: 24 blocks of street, walked in 28. A street network with its odd-degree vertices marked and the streets a shortest closed route must walk twice drawn doubled, dashed in a second colour, pairing up the odd vertices.

    The streets a postman walks twice

    A postman must walk every street of a district and come back. If every corner has an even number of streets, no street needs walking twice. If not, some must — and the ones repeated always join the odd corners in pairs. Pricing every way of pairing them finds the shortest round; pairing the nearest corners first does not.

    part 3 · discrete

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