Preface
I am currently preparing for applied mathematics and informatics olympiad qualifiers in my university, and while preparing I have stumbled upon the following task, which is quite hard for me to deal with.
The problem
How many strict undirected graphs with $8$ vertices each of degree $5$ are there (up to algebraic isomorphism)? Draw at least one of them, showing if it is planar or not.
My attempt
I was able to create exactly $3$ graphs of the above construction, knowing that any graph of such construction has $|E|=\frac{8\cdot 5}{2}=20$, and have shown that each of them has a subgraph homeomorphic to $K_{3,3}$ (so none of them are planar).
Below I have attached the image of these three graphs with red edges showing a $K_{3, 3}$-homeomorphic subgraph for each of the graphs. Yet, I cannot think of any other graphs, which would follow the problem's proposed structure. As it seems to me, each of these graphs have their edges starting from any vertex iteratively shifted from closest to farthest vertices (I do not really know how to describe it in better words). It also seems like each of the graphs has a Hamiltonian path.
The answer for the first part of the problem (number of such graphs possible) is $3$. It seems to comply with the fact that I cannot come up with any other graph constructions except the ones I have already created. Yet, I cannot devise any formal solution to prove that there are no more than $3$ such graphs. And so I wonder: is there any formal way to show this?
Any help will be appreciated, thank you in advance!