Comments for Short, Fat Matrices
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a research blog by Dustin G. Mixon
Thu, 13 Dec 2018 22:11:54 +0000
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by K
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-9198
Thu, 13 Dec 2018 22:11:54 +0000http://dustingmixon.wordpress.com/?p=4933#comment-9198I am following this progress as a not-graph-theory-expert, and find it quite fascinating. I enjoy very much reading the overviews in the thread, which I can follow (or which motivates me to look into certain concepts). There has not been a new overview since septmber. Is one planned at some point? I am not really able to follow the progress on a big picture, for instance – what is the smallest graph that is not 4-colorable? What are the understandings of these sort of graphs? What is the property that makes it 4-colorable? Is the progress of this polymath large enough to merit a publication at some point? Thank you so much for doing the polymath, which allows others (like me) to follow live research ðŸ™‚
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by domotorp
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-8097
Mon, 26 Nov 2018 19:56:37 +0000http://dustingmixon.wordpress.com/?p=4933#comment-8097We only have a proof in the plane with several restrictions, now on the sphere I would like to have an example without any restrictions (this is what I call conjecture), similarly to the 24 vertex graph in the plane.
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by ag24ag24
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-8051
Mon, 26 Nov 2018 13:11:06 +0000http://dustingmixon.wordpress.com/?p=4933#comment-8051Yep. (Let’s call the special vertex the south pole, though, not the origin, so that people don’t jump to the conclusion that it is the centre of the sphere.) We got down to 26 vertices for this without the half-plane restriction, right? And hang on, I thought you had a proof for the panar version? – what am I forgetting?
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by domotorp
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-8019
Mon, 26 Nov 2018 08:36:56 +0000http://dustingmixon.wordpress.com/?p=4933#comment-8019If we anyhow assume that the bichromatic origin conjecture is true, then we might as well assume that it’s also true on the sphere. Hopefully this would significantly reduce the size of a 5-chromatic graph, just like in the planar case. So the new task would be: Is there a unit-distance graph on the surface of the unit sphere such that we cannot 4-color it if the origin (a given special vertex) needs to be given two colors?
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by domotorp
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7140
Thu, 15 Nov 2018 09:02:16 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7140Thanks. Indeed it’s hard to see a way to get around the fact that it might be the clamp that becomes 6-colored.
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by ag24ag24
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7130
Wed, 14 Nov 2018 20:59:03 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7130Ah, sorry – check back for my description of a 59-vertex 6-chromatic graph in R^3. Basically I take two copies of Nechushtan’s (2000) “basic construction” and spindle them so that the two copies of his flap (the thing that he rotates to get his torus) can have their tips merged and still be mobile relative to the main structure. That 20-vertex thing, which I call a clamp, then has two vertices whose separation can vary between about 1 and 2.5 and that must be different colours in any 5-colouring. Thus, any graph with six vertices all pairs of which are between about 1 and 2.5 apart can be “clamped” – each pair not exactly 1 apart being identified with those two vertices of a clamp – and the result is a graph in which either the six-vertex “scaffold” uses six colours or else one of the clamps does (because its clamping vertex pair are the same colour). The obvious scaffold is of course the unit-edge octahedron, which requires just three clamps, giving a total of 60 vertices. (Then rotation can lose one of them, but that’s a boring detail.)
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by domotorp
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7128
Wed, 14 Nov 2018 20:28:14 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7128Could you please remind me what these clamps are?
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by ag24ag24
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7126
Wed, 14 Nov 2018 18:11:16 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7126Also, there may be scope for building something out of multiple unit-radius spheres with centres 1 apart. I’m keeping in mind the potential to use my 20-vertex clamps to enforce 6-chromaticity of subgraphs, though as yet I am not seeing a good way to use them (since they only enforce that either one of the clamps or the clamped thing must use six colours).
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by ag24ag24
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7123
Wed, 14 Nov 2018 16:11:00 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7123PS: an efficient way to add vertices in quite edge-dense configurations seems to be to reflect the entire structure in a hexagon. This introduces new hexagons and can thus be iterated. I’m somewhat optimistic that this might lead to structures whose 4-colourings all have a property that invites spindling.
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Comment on Polymath16, eleventh thread: Chromatic numbers of planar sets by ag24ag24
https://dustingmixon.wordpress.com/2018/09/14/polymath16-eleventh-thread-chromatic-numbers-of-planar-sets/#comment-7122
Wed, 14 Nov 2018 15:15:52 +0000http://dustingmixon.wordpress.com/?p=4933#comment-7122Excellent. I’ve been travelling and unable to make time to look at this other than in my head, but there does seem to be quite a lot to try. One useful fact is that in a 4-colouring of the cuboctahedron no hexagon can be coloured with just two colours (“2tri” as we were calling it months ago) because that would force the two triangles to use three other colours. I’ve been trying to get stronger results, like eliminating 1tri by combining multiple cuboctahedra, but without success so far.
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