Subcubic graph number

The subcubic graph numbers are the outputs of an extremely fast-growing combinatorial function. They were devised by Harvey Friedman.

A subcubic graph is a graph in which each vertex has a valence of at most three. (For the sake of this article, subcubic graphs are allowed to be s or s, and are not required to be connected.) Suppose we have a sequence of subcubic graphs G1, G2, ... such that each graph Gi has at most i + k vertices (for some integer k) and for no i < j is Gi into Gj.

Friedman showed that such a sequence cannot be infinite, and for each value of k, there is a sequence with maximal length. We denote this maximal length using SCG(k).

It is possible to show that SCG(0) = 6. The first graph is one vertex with a loop, the second is two vertices connected by a single edge, and the next four graphs consist of 3, 2, 1, and 0 unconnected vertices. All maximal sequences will peak and decline this way.

The value SCG(13) is the subject of extensive research due to its size. Not much is known about it, although it is far larger than TREE(3). Chris Bird suggested that it is larger than meameamealokkapoowa oompa, which is famous for being one of the largest named numbers.

The growth rate of SCG(n) is about \(\psi_{\Omega_1}(\Omega_\omega)\) in the fast-growing hierarchy.

If we require the subcubic graphs to be simple (i.e. not pseudographs or multigraphs), we get the simple subcubic graph numbers, denoted SSCG. Adam P. Goucher has shown that SSCG(2) << TREE(3) << SSCG(3). Moreover, he has shown that even TREEn(3) for any reasonable value of n is unnoticeable compared to SSCG(3).

Also, it can be shown that SCG(n) and SSCG(n) has comparable growth rate and Goucher has proved that \(SSCG(4n+3) \geq SCG(n)\).