Knigsberg Try to find a way to travel

Königsberg

Try to find a way to travel each bridge exactly once!

Problem of the Königsberg bridges *People believed that the task was impossible! *Euler solved Problem of the Königsberg bridges. *He wrote an article and published it in 1736.

Leonhard Euler • Born: 15 April 1707 in Basel, Switzerland • Died: 18 Sept 1783 in St Petersburg, Russia • The advisor is Johann Bernoulli.

Pictures of Euler

Commemorate

Graphs A graph G=(V, E) is an ordered pair with V, called the vertexset, and E, called the edge-set, where V and E are finite sets and the terminal nodes of each edge are vertices in V.

Definitions • A walk of a graph G is an sequence (v 1, e 1, v 2, e 2, …, vk) where v 1, v 2, …, vk are vertices of G and e 1, e 2, . . ek-1 are edges with the terminal vertices of ei being vi and vi+1. • A path of G is a walk with all vertices being different. • A graph is connected if for every two distinct vertices u and v, there is a path from u to v.

Definitions • A graph G with m edge has a eulerian circuit if G contains a walk (v 1, e 1, v 2, e 2, …, vm+1) with E(G)={e 1, e 2, …, em} and v 1=vm+1. • A graph G is called eulerian if it has a Eulerian circuit. • A eulerian path (or eulerian trail) of a graph G with m edges is a walk (v 1, e 1, v 2, e 2, …, vm+1) with E(G )={e 1, e 2, …, em}.

Definitions • A cycle is a walk (v 1, e 1, v 2, e 2, …, vk) with v 1, v 2, …, vk-1 are distinct and v 1=vk. • The degree deg(v) of a vertex v in G is the number of edges with v being a terminal vertex of these edges.

Königsberg bridges

Examples

Theorem A connected graph is eulerian if and only if, for every v V(G), deg(v) is even.

How to prove it? • It is easy to see that if G is eulerian then deg(v) is even for each v in V. • Conversely, suppose G is a connected graph with deg(v) is even for each v in V. Then we can find a cycle C in G. • Consider G-C, by induction on |E|, we can find that each connected part of G-C is eulerian. Combine C and eulerian subgraphs.

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