User blog:Hyp cos/TREE into SCG without diamond graph minor

Define SCGD(n) as the maximal length of sequence of graphs (G1, G2, ..., Gm) such that SCGD function grows much slower than SCG function, but not slower than TREE function.
 * 1) The degree of every vertex of every Gi is at most 3.
 * 2) For all i, Gi has at most n+i vertices.
 * 3) For all i < j, Gi is not graph minor of Gj.
 * 4) For all i, the diamond graph (K4 with one edge removal) is not graph minor of Gi.

Encode colored trees into SCGD
Here I'll show encoding of n-colored trees (the objects of TREE function) into subcubic graphs without diamond graph as minor (the objects of SCGD function).

The root (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)

A non-root node (colored m) with k children is encoded into(if it has less than 2 children, still let k=2)

To decode, first find the cycle system with 3 cycles, which is decoded into the root. Between two parallel-edge-pairs, there are two "paths", one is a 2-path (the color path), the other is a ≥3-path (the children path). The intermidiate nodes of the children path link to what the children of the decoded root are encoded into. For a cycle system with 2 cycles, which is decoded into a non-root node, find the node of cycle linking to its "parent cycle system". Between the node and the parallel-edge-pair, there are two "paths", one is a 2-path (the color path), the other is a ≥3-path (the children path). The intermidiate nodes of the children path link to what the children of the decoded node are encoded into. Finally, to determine the color of a decoded node, look at the subgraph the only intermidiate node of the color path links to. if there is a m-path between the intermidiate node and the degree-3 node, then the decoded node has color m; if there is no degree-3 node in the subgraph, which is an (n+1)-path, then the decoded node has color n.

Bound of SCGD
See here. The growth rate of SCGD function is at least \(\psi(\Omega^{\Omega^{\omega^2}})\).

However, the upper bound of growth rate of SCGD function is still unknown, except the limit from SCG function - \(\psi(\Omega_\omega^\omega)\).