User blog comment:Wythagoras/Extension of SCG/@comment-1605058-20130813135100/@comment-5529393-20130814112823

I think CSCG(0,4) is  very large.

We can start:

G1: double loop

G2: double edge

G3: vertex connected to a loop and two edges

G4: vertex connected to a loop and a 3-path

G5: vertex connected to a loop and a 2-path

G6: vertex connected to a loop and an edge

G7: loop

G8: tree with two vertices of valence 4

G9: vertex connected to four 2-paths

From G9 on we can have a vertex with four edges, each edge connected to a different tree. So we will write it as (a,b,c,d) where a,b,c,d are the four trees.

G9: (edge, edge, edge, edge)

G10: (0, edge, 2 edges, 2 edges)

G11: (0, 2-path, 2-path, 2 edges)

G12: (0, 2-path, 2-path, 3-path)

G13: (0, edge, 5-path, 2 edges)

G14: (0, edge, 4-path, 2 2-paths)

G15: (0, edge, 4-path, (((x)x)x) )

The fourth tree continues the sequence for tree(3), which we determined to be of length at least 2^18-4. We then continue to (0, edge, 3-path, binary tree with 2^18 edges). So we can continue for something on the order of $$F_{epsilon_0 * 2} (F_{epsilon_0}^3 (18) $$ graphs. Then we go to subcubic trees, for which it's not clear how fast it grows.