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.