# Help me find the shape

Hey guys.

I am trying to find a shape that suits my needs.

The amount of vertexes can be any, but the fewer the better.

The edges are special in that each end has an A or a B. like this A------B

I need the vertexes to have more A than B inputs from edges.

I need all vertexes to have the same amount of edges.

I have tried various 3dimensional shapes to no avail. Maybe someone here can help me. I have even tried shapes inside of shapes.

A-------A

If each edge needs to have an A and a B then unless you can connect two edges together like A-------BB-------A I don't see how you can take an equal number of As and Bs and have more As than Bs at your vertices.

ya, that's the way it's looking.

Keep in mind this can be in any dimension. 2, 3, 4,5. just as long as I get more vertexes with more A than B.

A-----A could be possible, assuming the end result was still the same. Same with
B-----B.

each vertex should look like
Let n < m
Axn
Bxm

technically it could have
A----A
A----B
B----B
B----A
or even
/-1/2B
a--
\--1/2B
But the last one would make things a real mess.

what ever method is used, though, the same ratio and values of A:B has to be on each vertex.

edit: right now I am looking at 2-d shapes. before I was looking at 3d.

edit2: in 2d I have found the pattern of vertexes = A +1, where B = 2A. for example 4 vertexes with one a and 2 b. This works when the edges are allowed to change. I remember last night dreaming something about repeating decimals, so now I have to figure out what that meant. I think the lower the repeating decimal the better? but I could be wrong.

Edit 3: I just realized I wrote out that pattern wrong. # of vertexes = A+1+B

Last edited:
Doh!... Wasn't thinking.

The issue comes up that one A is actually existing as two. So over all the vertex would still have A=B. which doesn't work.

Back to the drawing board.

Anyone know of geometry software that could try to calculate that?

I have noticed that Cayley Graphs allow for Arrows, but only allow one going out and one going in. Does anyone here know of a theory that allows for set number of out arrows and set number of in arrows?

I don't have a solution but these definitions or terms may be helpful:

- If every edge connects exactly one vertex A and one vertex B this means your graph is 2-colorable.

- For an undirected graph, i.e. edges don't have arrows, the number of edges incident to a vertex is called degree.

- For a directed graph (edges have arrows):
The number of edges going out of a vertex is called outdegree.
The number of edges pointing to a vertex is called indegree.

Maybe you can look for a theorem using these terms.