# Message #2250

From: Melinda Green <melinda@superliminal.com>

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

> <http://en.wikipedia.org/wiki/Gr%C3%B6tzsch_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!

-Melinda