Prove Corollary 10.1.5.

Prove Corollary 10.2.5.

Given information:

G is a graph

v and w are two distinct vertices of G.

Proof:

First part:

Let ve1v1e2v2...en−1vn−1enw be an Euler path from v to w. this then implies that all edges in G are e1,e2,...,en−1,en, while the Euler path also contains all vertices of G at least once.

Let us define a (new) edge e from w to v and let G′=G∪{e} (which is the graph G with an added edge). ve1v1e2v2...en−1vn−1enwev is then an Euler circuit in G′.

A graph contains an Euler circuit if and only if the graph is connected and all vertices of the graph have a positive even degree. This then implies that G′ is connected and all vertices of G′ have an even degree.

G is then connected, because G is G′ without the edge e (even though we removed an edge between v and w, ve1v1e2v2...en−1vn−1enw is still a walk from v to w )