I'm working on a problem in logic and it reduces to proving that a certain infinite graph is connected. The graph has the following properties:
- It is bipartite
- It is not necessarily finite (or countable or locally finite).
- It has no degree zero vertices
- It has no infinite rays
- It is acylic
My question is a bit open-ended. I am finding it difficult to prove the connectedness. I was wondering if there are any necessary and sufficient conditions to ensure connectedness. In other words, if I could try proving a property on graph that is equivalent to its connectedness.