User blog comment:Wythagoras/Extension of SCG/@comment-1605058-20130813135100/@comment-1605058-20130815082406

Let graphs up to G9 be as in Deedlit's sequence. Using same notation as he does, here is good bound with a lot places for improvement (lazy me):

G10: (0,edge,edge,edge and 2path)

G11: (0,edge,edge,2-path with two edges at the end)

G12: (0,edge,edge,edge with two edges at the end)

G13: (0,edge,edge,two edges)

G14: (0,edge,edge,7-path)

G20: (0,edge,edge,edge)

G21: (0,0,edge,7-path and 8-path)

Now I'll only look at tree on the left. It will have one valence 3 node, a vertices between this node and valence 4 node, and b-path and c-path coming from it. This will be denoted (a,b,c).

G21: (0,7,8)

G22: (2,7,7)

G24: (0,7,7)

G25: (0,6,13)

G26: (2,6,12)

G28: (0,6,12)

G29: (6,6,11)

G about 2^8: (0,6,6)

G one later: (0,5,2^8)

Note that while b and c are interchangable, a is independent. By later deleting an edge from G21 and continuing I think we get to about 2^^2^^8.