Definition fundamental cycle: If $T$ is a spanning tree of a given graph $G$, and $e$ is an edge that does not belong to $T$, then the fundamental cycle $C_{e}$ defined by $e$ is the simple cycle consisting of $e$ together with the path in $T$ connecting the endpoints of $e$.
I feel like this should be done as a proof via contradiction, but I also feel like my solution is wrong:
Proof by contradiction, assumption: The edge does not lie on the fundamental Cycle in the spanning tree of $G$.
Then there are two vertices $a,b$ in $T$, that are connected by 3 Paths:
- The fundamental path, this is the only path in T that connects $a,b$ and always exists in a tree.
- The path $a,b$, which is the edge $\{a,b\}$ ($e$)
- Another path that is not the fundamental path nor the path $a,b$
But there is only one path in a Tree that connects two vertices, the fundamental path and only one edge that directly connects $a$ and $b$, the edge e. Therefore there can not be another path. Because both, the fundamental path and e lie on the fundamental cycle we have our contradiction.