This is more of a general question that arises from the following:
I was asked to find all nonisomorphic cubic graphs (graphs in which every vertex has degree $3$) with $4$, $6$, and $8$ vertices. The way I tackle this is to start "coming up" with graphs of $p=4,6,8$ vertices without too much logical thought. I then check that the graphs of $p$ vertices are not isomorphic to one another by labelling the vertices of some pair of graphs and trying to define a mapping $f$ that would make $f$ an isomorphism.
How would I approach such a problem more efficiently?
Specifically, I want to show that two graphs $G_1, G_2$ are not isomorphic by showing that there is no bijection $f:V(G_1)\to V(G_2)$ that preserves adjacency. With the basic knowledge I have, I'm simply trying to relabel vertices via $f$. This would look something like what I see in my textbook:
Thanks in advance for the help!