User blog:Wythagoras/Extension of SCG

This blog post is about an extension of SCG. I define SCG(a,b) as the maximal length of a sequence of graphs where the graphs has a valence of at most b. The nth graph has at most a+n vertices. We have to go to a regular minor for valence greater than 3.

Deafult cases
SCG(a,0) is a deafult case. We are only allowed to use dots. For SCG, SSCG, PSCG and PSSCG this will be a sequence of length a+2, for CSCG, CSSCG, PCSCG and PCSSCG is it only 2 beacause we are not allowed to use 2 or more dots.

SCG(a,1) is a deafult case too. We are only allowed to use dots and sticks.

SCG(0,1) = 2: dot and empty

SCG(1,1) = 5: stick, 3 dots, 2 dots, 1 dot and empty

SCG(2,1) = 8: stick+dot, stick, 5 dots, 4 dots, 3 dots, 2 dots, 1 dot and empty

SCG(3,1) = 13: 2 sticks, stick+3 dots, stick+2 dots, stick+1 dot, stick, 8 dots, 7 dots ... 1 dot and empty

In general: SCG(n,1) = \(F_{n+4}\) where \(F_{n}\) is the nth fibonnacci number

For CSCG and variants is it equal to 3: stick, dot and empty.

SSCG(a,2) is deafult case: we can only use path forests. therefore \(SSCG(a,2) \approx f_{\omega}(a)\)

For CSSCG is it only n+2.

valence and type.