## Graph Theory Lecture Notes10

### Königsberg Bridge Problem

Kaliningrad, Russia (Just north of Poland).

Späziergang - a walk around town usually on Sunday afternoon.

Kneiphoff - the island in the Pregel River in Königsberg.

**Def**: An *eulerian closed chain* is a closed chain in a multigraph which uses each edge exactly once.

**Theorem** (Euler, 1736): A multigraph G has an eulerian closed chain if and only if G is connected
up to isolated vertices and every vertex of G has even degree.

### An Algorithm for Finding Eulerian Closed Chains

Arbitrarily select edges forming a chain until the chain closes at the starting vertex. If there are
unused edges, go to a vertex that has an unused edge which is on the chain already formed, then
create a "detour" - select unused edges forming a new chain until this closes at the vertex you started
with. Expand the original chain by following it to the detour vertex, then follow the detour, then
continue with the original chain. If more unused edges are present, repeat this detour construction.

### Further Results on Eulerian Chains and Paths

**Theorem** (Euler): A multigraph G has an eulerian chain if and only if G is connected up to isolated
vertices and the number of vertices of odd degree is either 0 or 2.
**Theorem** (Good): A multidigraph D has an eulerian closed path if and only if D is weakly connected
up to isolated vertices and for every vertex, indegree equals outdegree.

**Theorem** (Good): A multidigraph D has an eulerian path if and only if D is weakly connected up to
isolated vertices and for all vertices with the possible exception of two, indegree equals outdegree,
and for at most two vertices, indegree and outdegree differ by one.