# Prove that the property is preserved under isomorphism.

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?

To illustrate, consider the degree $$4$$ vertex $$u$$. Since it has degree $$4$$, there are four vertices attached to it by edges, say those vertices are $$v_1,v_2,v_3,v_4$$. Then because $$uv_1$$ is an edge, and $$f$$ is an isomorphism, then $$f(uv_1)$$ is an edge between $$f(u)$$ and $$f(v_1)$$.
This argument is not complete yet. How do we know $$f(u)$$ is degree $$4$$ exactly? That is, it doesn't have more/less adjacent vertices than we expect.