I've run into a problem with the definition of a tree, and possibly more generally with the definition of a cycle. I've run into the problem a few sections after we talked about trees, and I never really thought of the problem until I got into Hamiltonian cycles. I will state my problem in three steps: 1. Fact: the complete graph on 2 vertices is Hamiltonian and so contains a cycle. 2. Fact: a tree is a connected acyclic graph. 3. Therefore, since every tree besides the trivial tree contains K2 as a subgraph, there exist no trees.
Now, obviously something is missing in the details. My book clearly states the definition of a tree as a connected acyclic graph. So, is it not the case that K2 is Hamiltonian? Is there no such thing as a 2-cycle?
Is my definition of a tree not adequate? Does it exclude 2 cycles from the necessity of there not being a cycle in the graph?
Please, someone help me to clear this up.
Your definition of tree is fine. Partly to avoid this difficulty, by convention $K_2$ is not considered to be Hamiltonian.