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

Question

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?

Answer 1

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.

Answer 2

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