User blog comment:LittlePeng9/SCG(1)/@comment-5529393-20130313042828

Major improvement!

I will ignore forests for the moment, so that we get a provable lower bound. Let's start with a vertex with a loop connected to a subcubic tree. We can consider this as a rooted binary tree,  so it grows at the rate of at least F_epsilon_0 (n). Next, consider a 2-circle with a path of length n sticking out of one vertex. Each time we remove an edge from the path, we can add a graph consisting of a vertex with a loop connected to a maximal subcubic tree, i.e. a maximal binary rooted tree. So we diagonalize over the previous function F_epsilon_0 (n), i.e. we get F_{epsilon_0 + 1} (n). Next consider a 2-circle with a tree sticking out of one end. The tree is:

The root has one child, vertex A.

A has two children, vertices B and C.

B has a path of length n descending from it; C is childless.

Each time we we remove an edge from the path descending from B, we can add a maximal path between the root and A. This allows us to diagonalize over the previous function F_{epsilon_0 + 1} (n), leading us to F_{epsilon_0 + 2} (n).

We continue in this fashion through higher and higher-valued trees. Since binary rooted trees have ordered type epsilon_0, we get to F_{epsilon_0 * 2} (n).

That is just 2-circles with a tree sticking out of just one vertex. Now we deal with trees sticking out of both vertices. Let's start with one edge sticking out of one vertex and a path of length n sticking out of the other. Each time you remove an edge from the n-path, you can add a 2-circle with a tree sticking out of just one side. So we diagonallize over the previous function F_{epsilon_0 * 2} (n), leading us to  F_{epsilon_0 * 2 + 1} (n). Continuing in this fashion, a 2-circle with an edge sticking out of one side and a general subcubic tree sticking out of the other will have growth rate  F_{epsilon_0 * 3} (n). Next, going to a path of length two sticking out of one vertex and a general subcubic tree sticking out of the other will get us to  F_{epsilon_0 * 4} (n). Continuing, we eventually get that a 2-circle with general subcubic trees sticking out of both sides will have growth rate  F_{epsilon_0 ^ 2} (n).

Next, we go to 3-circles. By an argument similar to the ones we have made before, a 3-circle with subcubic trees sticking out of all three sides will get us to  F_{epsilon_0 ^ 3} (n). More generally, an m-circle with subcubic trees sticking out of all m sides will get us to  F_{epsilon_0 ^ m} (n). Diagonalizing over m-circles will get us to  F_{epsilon_0 ^ omega} (n).

Okay, now we are ready to get a lower bound for SCG(1). We start with

G2: two vertices with three edges between them

G3: vertices A, B, C, where there are two edges between A and B, an edge between B and C, and a loop at C.

G4: vertices A, B, C, D, with edges from A to B, C and D, and loops at B, C, and D.

G5: vertices A, B, C, D, E, with edges from A to B, C, and D, loops at B and C, and an edge from D to E.

G6: graph H  plus a 2-circle, where

H is vertices A, B, C, D, with edges from A to B, C, and D, and loops at B and C.

G7: graph H plus three isolated loops.

G8: graph H plus two isolated loops and an isolated edge.

G9: graph H plus two isolated loops and three isolated vertices.

G10: graph H plus two isolated loops and two isolated vertices.

G11: graph H plus two isolated loops and one isolated vertex.

G12: graph H plus two isolated loops.

G13: graph H plus one isolated loop and a maximal tree with 8 vertices.

This continues for tree(8) steps, until we get to

G tree(8)+: graph H plus one isolated loop

G tree(8)+: graph H plus a maximal tree with tree(8) vertices.

...

G tree(tree(8))+: graph H

G tree(tree(8))+: vertices A, B, C with edges from A to B and C, and loops at B and C, plus a circle of size tree(tree(8))

We define circle(n) to be the longest possible sequence starting from a circle of size n. Note that circle(n) has growth rate F_{epsilon_0 ^ omega} (n).

G circle(tree(tree(8)))+: vertices A, B, C with edges from A to B and C, and loops at B and C.

G circle(tree(tree(8)))+: vertices A and B with an edge from A to B and loops at A and B, plus a circle of size circle(tree(tree(8))).

G circle(circle(tree(tree(8))))+: vertices A and B with an edge from A to B and loops at A and B.

G circle(circle(tree(tree(8))))+: a circle of size circle(circle(tree(tree(8)))).

This finally leads to a sequence of at least circle(circle(circle(tree(tree(8))))) steps.

So SCG(1) > circle(circle(circle(tree(tree(8))))), where circle grows at least as fast as  F_{epsilon_0^omega}(n), and tree is conjectured to grow at least as fast as F_{epsilon_0}(n).