Prove that the following properties: ‘There exists a vertex of degree 4’ and ‘There exists a cycle of length 4’ are preserved under isomorphism.
By definition, a property of a graph is said to be preserved under isomorphism if whenever G has that property, every graph isomorphic to G also has that property.
I know how reason about the first property, for example: since an isomorphism is a bijection between sets of vertices, isomorphic graphs must have the same number of vertices. If one graph has a vertex of degree 4, then another graph should also have a vertex of the same degree so they can be isomorphic. Thus, $v$ and $f(v)$ will have the same degree.
Would it be sufficient to say this for the first? I am not sure how to construct a formal proof for these properties. How can one prove the second property?