Let $T$ be a spaning tree of a connected graph $G$ and let $e$ be an edge of $G$ not in $T$.Show that $T+e$ contains a unique cycle.
So we know that if $T$ is spanning tree $\rightarrow$ $T$ is maximally acyclic, which means adding any edge will create a cycle, $\rightarrow$ $T+e$ creates a cycle. We can assume by contradiction that $T+e$ is not unique cycle which means that $\exists$ another cycle which is a contradiction, since spanning tree $T$ does not have any loops or cycles, since removing one edge from spanning tree will make the graph disconnected, since spanning tree is minimally connected.
I don't know if this is enough to say that i "solved" this problem, maybe i should add something more?
Thanks in advance.
Best Answer
not exactly, but close. you need to prove e forms a cycle as well. It is pretty simple, but it is necessary. (also T+e contains a unique cycle, not is a unique cycle)