We know the common result : - If every vertex of a graph G has degree at least2, then G contains a cycle. Can I conclude that 2-regular graphs are cycles where degree is exactly two of every vertex? I am not getting any contradictory example. Please help me if am wrong. Need some help.
I worked like this: If the graph G has n vertices of degree two. Let us assume that all vertices have degree two except two vertices, say vertex v and u. now to make degree two of vertex v and u, we must attach them with other vertices. only possible case is to make vertices u and v together. otherwise if we make them adjacent to some other vertex, then degree of that vertex will be three or more. Contradiction. Is my conclusion right?
NOTE: $G$ is a connected graph.