Create a configuration - graph theory

Question

I've encountered this (startling) difficult, to me, question:

Create a configuration in the plane with a ring size 4, so that every internal vertex is of degree 5.

Now, I assume I may not use multiple edges - or it wouldn't be a real configuration. Am I mistaken?

Furthermore, trying to insert a pentagon or anything of that sort just doesn't work. Anything with a square inside is rather futile, as you end up trying to create vertices inside it with degree 5 - back to the original problem. Trying to 'correct' the issues with another vertex only makes it far more complicated. It seems as if I am missing something crucial in here, perhaps in the definitions themselves, or in some general idea as to how to create such a shape. Can someone guide me to the right shape?

Any hints/direction will be extremely appreciated!

Answer 1

I'm assuming that you want to construct a planar graph and its embedding such that

• the graph contains $C_4$ cycle as a subgraph,
• in the given embedding all the vertices that happens to be inside the area enclosed by that (some particular) cycle have $\deg(v) = 5$.

Hint:

• The icosahedron graph is a 5-regular planar graph (see here).

I hope this helps $\ddot\smile$