So, a vertex is called a leaf if it connected to only one edge.
a) Show that a tree with at least one edge has at least 2 leaves. b) Assume that G = (V, E) is a graph, V ≠ Ø, where every vertex has at least 2 edges, Show that G has a cycle.
I don't really know for sure how to write the proofs for these two tasks, but here is what I have
a) So between two points u, v in the graph there is always exactly one path, and if G is a tree it is connected. For a Graph with e=1 edges you will have n=e+1 vertices. Is there any more to it than this? Do I have to include something else?
b) So G is a connected graph iff G=(V,E{e}) has a cycle that contain {e}. Now, every node has the degree 2, meaning they are connected to vertices. This also means that the end vertex is connected to 2 vertices. Very simplified I have written
u,v ∈ V V = {u(n),......u(n+1), v} Now, if G did not contain any cycles that would be it, but since v is connected by 2 vertices it means it is reachable by another vertex than u(n+1) right? having a problem showing this in a sensible way.
Thanks in advance.