It is possible to to get from one node to any other node in the community through a series of the underground tunnels. If $k$ is the maximum number of tunnels that you can pass through in a row without visiting a node twice, prove that any two paths that go through $k$ tunnels without visiting the same node twice must pass through a common node.
I'm having trouble wrapping my head around this problem. I think that that common node should be in the middle of both the paths. Because if this is not the case then we can have a path of length >k, but am unsure of how to proceed.
Assume there are two such maximal path $p_1,p_2$ that don't intersect. Their sets of vertices are disjoint.
By the connectivity there must be some path $p_3$ joining some vertex $v_1$ of $p_1$ to a vertex $v_2$ of $p_2$. The path $p_3$ can be chosen to not share edges with $p_1$ and $p_2$. Now travel the first path until $v_1$ (there are two pieces to choose so choose the largest, which is at least half of $p_1$), follow $p_3$ and then travel the longest piece in which $v_2$ divides $p_2$ (which is at least half the length of $p_2$).
That new path has length $\geq k/2+|p_3|+k/2>k$, which is a contradiction with $k$ being the maximum length of a non-intersecting path.