As title, (note: this questions is asking weather or not all vertices are contained IN a cycle not asking if the G contains a cycle.
So this graph would be an counter example that disprove that every vertex is contained in a cycle. Would like to know if this is a valid counter-example since V1 is not contained in a cycle.
PS: As I was finding examples, also observed that if the all V have exactly degree of 2, then it is necessary that all vertices are contained in a cycle. Please prove if I am missing something.