The simplest $\Gamma_{\mathfrak B}$ graph where squares are separated by two hexagons... Let's call a bicubic planar graph build up of faces having degree $4$ and $6$ only, a $\Gamma_{\mathfrak B}$ graph. The simplest...--prophetes.ai

The simplest $\Gamma_{\mathfrak B}$ graph where squares are separated by two hexagons... Let's call a bicubic planar graph build up of faces having degree $4$ and $6$ only, a $\Gamma_{\mathfrak B}$ graph. The simplest one is the truncated octahedron. Its planar drawing looks like the following: ![enter image description here]( Obviously every square is separated by at least one hexagon. > What is the simplest $\Gamma_{\mathfrak B}$ graph where the squares are separated by at least two hexagons? I found a way to extend it by traversing it in the following way (the sharp turns of the outmost edge are due to Paint and not vertices of degree $2$): ![enter image description here]( But it get's messy when I continue traversing, so I thought there is a way to simply expand a hexagon. Any idea welcome...

The picture shows one example of 2 lines crossing the original graph and one round trip leaving all squares separated by at least 2 hexagons.

![example divided graph](