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Paths, cycles, trees, Euler and Hamilton

Connectivity, spanning trees, Euler circuits and the bridges of Königsberg.

A tree is a connected graph with no cycles: n vertices, n − 1 edges. Euler proved a graph has a circuit using every edge once exactly when every degree is even. The Königsberg bridges failed. Picture it: the seven bridges; four land masses with odd degree. Think it: Euler's condition is local (degrees) yet decides a global question, the first theorem of topology.

Worked example · 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

Step by step

  1. \binom{5}{2} = \frac{5!}{2!\,(5-2)!}

    Unordered selections: n! / (k! (n−k)!).

  2. = \frac{120}{2 \times 6} = 10

Reveal the answer
\binom{5}{2} = 10

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How to: Paths, cycles, trees, Euler and Hamilton

  1. Unordered selections: n! / (k! (n−k)!).

Questions people ask

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

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The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesColouring and planar graphs