I also got 625-vertex 5-chromatic graph with . It contains large (469-vertex) and small (157-vertex) subgraphs, connected via zero-weight vertices of large subgraph, so small subgraph does not affect FCN value. I think the same is true for Maijn’s 5-chromatic graphs too.

Vertex coordinates are available here:

https://www.dropbox.com/s/9jvu56xzbyykdci/g655.vtx?dl=0

https://www.dropbox.com/s/57x3aeubbheahz7/g625.vtx?dl=0

https://www.dropbox.com/s/xp7b43bnnrsqz0c/g469.vtx?dl=0

I also made two observations about this kind of proofs:

1. If in a graph associated with a set of five points we fix which edges have length 1 and which ones have length such that it gives us a lower bound for $p_d$, then this graph cannot be realized both for and for (at least with the present upper bounds).

Proof: These point sets only work if there are at least 8 segments from (if there would be only 7 such segments, the sum of the known upper bounds for the ‘s of the remaining edges would be at least 1). So there are at least 24 such triangle-side pairs, where the side is of length 1 or of length (there are 3 possible triangles for all such segment), so at least 4 triangles which have all of their side-lengths as one of these two (otherwise there could only be at most such pairs). But with going through the cases which only include equilateral triangles for these four (3 unit-length equilateral triangles and one length, 2 unit-length and 2 -length, …) we can easily see that this is impossible. So there is a triangle with side-lengths or , which implies the statement.

2. Similarly to the 5 point case, we could also use this method for larger point sets but if we only use the number of monochromatic edges, but not their location, then we can see that there are only finitely many such point sets which can give us lower bounds this way (because of the upper bound for the edge number of unit distance graphs). From this, it concludes that some small neighbourhood of 1 cannot be covered if we follow this method strictly (at least as of the current upper bounds). But the location of the monochromatic edges isn’t arbitrary (they are forming 4 cliques), so if we use this too, there might be an infinite set of graphs which covers everything but 1. I might write something more detailed about this later.

]]>I tried to process Marijn’s 553-vertex graph with no success. The progress is very slow to see the end.

I think here is the answer to your CN-FCN question (or not?):

https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-6731

It seems you are mighty close to being able to test Marijn’s smallest 5-chromatic graph. Do you think it is within reach?

I think we still need to think about a different question too, namely whether there is a UD graph that has CN-FCN greater than 1 (or even equal to 1). As far as I know, nothing has been found that beats the case of increasingly large odd cycles, for which CN=3 and FCN asymptotically approaches 2 from above. Do you have any insights?

]]>Now my FCN record is . The corresponding graph has 649 vertices, 3924 edges, 1020 spindles and vertex degree in range from 8 to 30.

The pictures of graphs are shown below (numbers from top to bottom are vertex number, vertex weight divided by 10000 and orbit number).

Vertex coordinates are available here:

https://www.dropbox.com/s/50qhqx3oq2d5vfm/g607.vtx?dl=0

https://www.dropbox.com/s/5sbkbizym1lu98o/g649a.vtx?dl=0

It is always possible to get a graph with higher FCN value, the problem is to calculate FCN. The computation time increases roughly 10 times for every 100 additional vertices. It also depends on “niceness” of graph, but I still don’t know what makes the graph nice. The next problem is restrictions of trial versions of solvers. Maximum order of the graph is about 650…700 for trial version of Gurobi.

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