Are 4-chromatic 3-connected unit distance graphs always rigid? 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.

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