# Existence of graphs of every order related to Barnette’s conjecture

Consider the class C3CBP of $$3$$-connected cubic bipartite planar graphs. They form the class on which the (in)famous Barnette’s conjecture is based. My interest in C3CBP graphs is somewhat orthogonal to this:

My questions are:

1. Does there exist a C3CBP graph on $$n$$ vertices for every even $$n > n_0$$ where $$n_0$$ is some small constant?
2. Does there exist a C3CBP graph without any non trivial automorphisms on $$n$$ vertices for every even $$n > n_1$$ where $$n_1$$ is some small constant?

Notice that $$3$$-connected planar graphs have only at most linear sized automorphism groups so the second question is natural.

I need the existence and explicit construction of such graphs in an algorithmic problem apparently unrelated to Barnette’s conjecture.

I have a partial solution, with constructions for the first question and for the second question when $$n = 2\ (\text{mod}\ 4)$$.
Here are constructions for $$n = 8$$ and $$n = 14$$. These generalise to constructions for $$n = 0\ (\text{mod}\ 4)$$ and $$n = 2\ (\text{mod}\ 4)$$, by taking any square, for example 1,2,3,6 in the second construction, and replacing it with a ladder like shown in the third image.
For the second question, the $$2\ (\text{mod}\ 4)$$ construction has no nontrivial automorphisms when $$n \geq 18$$, as we can recover the labeling. First find the added ladder and the pairs 1-2 and 3-6. To discern between them, note that the nodes 1 and 2 are not in any four-node cycle except in the ladder, while 1 and 6 are. Similarly 2 and 3 are not in a four-node cycle except in the ladder, while 1 and 6 are. Then you can uniquely discern 4 (only non-ladder node connected to 1), 5 (same for 6), 8 (only node connected to 3 and 5) and so on.