Message #2250

From: Melinda Green <>
Subject: Re: [MC4D] color graph
Date: Mon, 04 Jun 2012 20:49:07 -0700

On 6/4/2012 8:26 PM, Roice Nelson wrote:
> I like how you laid out the graph, and how it emphasizes the special
> role of the white face :) It’d also be cool to see a presentation of
> the graph without intersections. That should be possible (embedded in
> 3D), though I guess it wouldn’t be as nicely symmetrical as the one
> you made.

Any graph can be embedded in R3 without crossings. I believe that this
one can’t be embedded in R2 without crossings.

> I searched around a little (on Google, Wikipedia, and Wolfram Alpha)
> to see if there was a special name for this graph, or for the graph
> you get after removing the red nodes, but did not have any luck. An
> interesting close call to the latter was the Grotzsch graph
> <>.

How does one search for a graph? That sounds super-useful!

> For this tiling/coloring, we have 16 faces, 30 edges, and 15 vertices,
> giving a Euler Characteristic of 1 (projective plane). Seems like a
> lot of the asymmetrical colorings end up producing that topology.

What an interesting observation, Roice!