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At braced heptagon we determined a graph was rigid. That graph is unit-distance, 3-connected, and 4-chromatic. Here are some more graphs, all extremely constrained. I'm pretty sure they are all rigid graphs, but I'm not sure.

4-chromatic graphs

Are any of these graphs not rigid? Are all unit-distance, 3-connected, 4-chromatic graphs rigid?

Ed Pegg
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I get:

  • the UnitDistance graphs above are rigid
  • the H-O-Fish graph is not infinetesimally rigid, but it might be finitely rigid

Further, using Mathematica 12.1.0

  • it doesn't seem to know the Exool... graphs
  • the embedding I get for the O-40 graph is not unit-distance

I may have to update my version of Mathematica

WRSomsky
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    "the embedding I get for the O-40 graph is not unit-distance" -- I'll get that fixed. – Ed Pegg Dec 24 '20 at 20:43
  • the graphs are named after Exoo and Ismailescu: this is an uppercase i, not a lowercase L. That's probably why you're not getting them in Mathematica, not the version number; my Mathematica 12.0.0.0 gets them just fine. – Misha Lavrov Dec 25 '20 at 20:24
  • @MishaLavrov Thanks for the spelling correction. That's what was giving me trouble there. – WRSomsky Jan 07 '21 at 11:41
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I've programmed the first seven graphs into my new Shibuya repository of unit-distance and integral embeddings of graphs. Strictly speaking I've only programmed six, but {UnitDistance, {15, 1}} contains {UnitDistance, {12, 2}} as a subgraph and the latter is (as will be shown) rigid, so the former is also rigid.

Using the same least-parameters method that I used to prove the rigidity of the braced heptagon, I get the following results. All the rigid graphs are first-order rigid:

Graph Least parameters required Rigid?
{UnitDistance, {9, 3}} 1 yes
{UnitDistance, {12, 2}} 2 yes
Exoo–Ismailescu 17 2 yes
Exoo–Ismailescu 19 2+1 yes
Exoo–Ismailescu 21 2+1+1 no (1 DOF)
Hochberg–O'Donnell fish 3+1+1 no (1 DOF)

The notation $a+b+c+\cdots$ means that a subgraph was parametrised using $a$ parameters, then based on this another subgraph was parametrised using $b$ parameters and so on – the result of all these phases is the whole graph.

Exoo–Ismailescu 21 is not rigid because there are two parameters but only one constraint in the first phase; the second and third phases each have one parameter and one constraint, so they don't help with rigidity:

The reason for the "hodfish" graph not being rigid is similar, only the first phase has three parameters and two constraints:

You would have done well to check the number of edges of the two graphs shown to be non-rigid above. They fail the Laman condition (one fewer edge than $2v-3$ where $v$ is the number of vertices), which is necessary for a graph to be rigid.

Parcly Taxel
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  • I also found that O'Donnell 40 wasn't unit distance when I tried constructing it. Nice work on Exoo21 and Hodfish. Do either have positions where another unit edge could be added? – Ed Pegg Dec 25 '20 at 14:44
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    @EdPegg Just look for a pair of vertices whose length changes from less than to greater than $1$ as the graph moves, then add an edge there. But for either graph there should be no edge that can be added that doesn't form a triangle or degenerates the graph. – Parcly Taxel Dec 25 '20 at 15:26