Prove Corollary 10.1.5.
Prove Corollary 10.2.5.
G is a graph
v and w are two distinct vertices of G.
Let be an Euler path from v to w. this then implies that all edges in G are 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 (which is the graph G with an added edge). is then an Euler circuit in .
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 is connected and all vertices of have an even degree.
G is then connected, because G is without the edge e (even though we removed an edge between v and w, is still a walk from v to w )
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