I have a picture of a rectangle and a vertex off to the side. I'd like to know if, since a Euler cycle includes all edges of a graph exactly once, if a Euler cycle exists in this graph.
Couldn't the rectangle itself be the Euler cycle? I know Euler cycles exist in connected graphs only, but this graph isn't connected and it appears the rectangle can be used as the Euler cycle for the graph.
What am I missing here? Why can't this be a Euler cycle? Any help appreciated.
It really depends on what definition you go by. Some definitions require the graph to be connected (in which case your example is not an Euler cycle), some do not require that but just require all edges to be visited, in which case your example is correct.
Often the assumption of connectedness is not explicitly stated, and that can indeed lead to confusion.