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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.

Urugero: 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

ku

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

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

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

i
\binom{5}{2} = 10

Symbols used here

\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

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.

Kuri Gukoresha

in Combinatorics & Graph Theory

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