Bicolored triangulations of $S^2$ with certain conditions on degrees of vertices

Question

Finite sets $B,W\subset\mathbb N^2$ are given. Suppose $G=(V,E)$ is a triangulation of a two-dimensional sphere so that $V=V_b\sqcup V_w$. We say that $G$ is a $(B,W)$-triangulation if $(\deg_bv,\deg_wv)\in B$ for every $v\in V_b$ and $(\deg_wv,\deg_bv)\in W$ for every $v\in V_w$. Here, $\deg_wu=|N_u\cap V_w|$ and $\deg_bu=|N_u\cap V_b|$ where $N_u$ is a set of all vertices from $V$ incident to $u$.

Suppose $B,W$ are so that the set of $(B,W)$-triangulations is finite. Q0: How do I determine all of them up to graph isomorphism? When is this set finite?

What I've tried is basically a bunch of double counting. Call vertex $v\in V$ black, if $v\in V_b$, or white, if $v\in V_w$. Since $W,B$ are finite, it's possible to write down the finite set $S$ of all possible subgraphs of the form $v\cup N_v$ (induced from $G$, of course). For every subgraph $T\subset S_i\in S$ with both black and white vertices, it's posible to count the number of embeddings of $T$ to graphs from $S$. For example, one can choose an edge betwen white and black vertices as $T$. This, accompanied by $\sum s_i=|V|$, $\sum|S_i|s_i=2|E|+|V|$ and Euler theorem, produces a system of equations over $s_i$, the possible number of vertices of 'type' $S_i$, for every $S_i\in S$. By the way, it's often not enough, and in some cases I can't even bound $s_i$.

So, the more concrete questions, based on my approach, are.

Q1: Are there any other tools for generating (or, at least, bounding) all $(B,W)$-triangulations?

Q2: Suppose I've bounded all possible $s_i$. That is, I know the number of vertices of each type. What is an effective way to find all the $(B,W)$-triangulations based on $s_i$?

The original problem is to find all $(B,W)$-triangulations for $B=\{(0,4),(3,2)\}$ and $W=\{(5,0),(4,2),(1,6),(2,5)\}$. I'll appreciate the hints for this particular problem, since my approach produces such a mess.

Answer 1

I don't have an answer to your question, but I have a couple of simple observations to find triangulations of the plane having $BW$-partition where $B=\{(0,4),(3,2)\}$ and $W=\{(5,0),(4,2),(1,6),(2,5)\}$.

Let $G=(V,E)$ be $BW$-triangulation of the plane:

1. If $v\in V$, then $\deg(v)=4,5,6 \text{ or }7$ (it follows from the fact that $\deg_d(v)+\deg_w(v)=\deg(v)$);

2. If $\deg(v)=4$, then $v\in V_b$ and $N(v)\subset V_w$ ($(0,4)\in B$);

3. Hence the set of all vertices of degree $4$ is an independent set in $G$;

4. If $v\in V$, $\deg(v)=5$, and $v$ is adjacent to a vertex of degree $4$, then $v\in V_w$ (follows from 2);

5. On the other hand, if $\deg(v)=5$ and $v\in V_w$, then $N(v)\subset V_w$ (since $(\deg_w(v),\deg_d(v))=(5,0)$);

6. It follows that no vertex of degree $5$ is adjacent to any vertex of degree $4$ and

7. the induced subgraph on the set $V_5$ of vertices of degree $5$ has at least two connectivity components of which at least one is a cubic graph;

8. If $\deg(v)=6$ or $7$, then $v\in V_w$;

9. $|V_b|\geq6$, $|V_w|\geq 6$ hence $|V|\geq12$.

Using the Plantri programme I checked that among the triangulations of the plane with at most $20$ vertices there are no triangulations that satisfy conditions 1,3,6,7,9.