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Graph-theoretically, a map is a cubic graph ${\displaystyle \Gamma }$ with edges colored blue, yellow, red such that: $\Gamma$ is connected, every vertex is incident to one edge of each color, and cycles of edges not colored blue, have length 4.

So I wonder whether graphs $\Gamma_\mathfrak B$ with the following properties

  • bicubic
  • planar
  • with faces having 4 and 6 vertices only
  • where the squares are separated (this is class 4 from this post here)

can represents maps?

Planarity would guarantee a 3-edge coloring via the 4-color theorem. I checked the planar drawing below which I thought it should be the truncated octahedron and it didn't work. Reading further on Wikipedia:

Note that $\Gamma$ is the flag graph or graph encoded map (GEM) of the map.

My question rephrased is: When are graphs $\Gamma_\mathfrak B$ gems?

EDIT Looks like I've taken the wrong graph enter image description here which is not the truncated octaeder, but a chamfered cube.

draks ...
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1 Answers1

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Every vertex must be in at least one square (since the red and yellow edges at the vertex must be part of a square), and it must be in exactly one square and two hexagons since you have specified that the squares are not adjacent.

So, we can build a cubic planar graph putting one square and two hexagons at each vertex; this forces us to make the graph of the truncated octahedron, so that's the only one.

Nick Matteo
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