# Combinatorial properties of this class of big girth, cubic graphs

Theorem (source, theorem 3.2) Given a simple graph $$H$$ on $$2^g$$ nodes consisting of disjoint cycles of girth at least $$g$$, one can add edges to make a new graph $$G$$ that is cubic (3-regular) and the girth of $$G$$ is at least $$g$$.

Tldr: the way it works is that it consider $$A \in (H \times H)\backslash E(H)$$ that has the desired property (aka $$(H, E(H)\cup A)$$ has nodes of degree $$\le 3$$ and of girth $$\ge g$$), and figures that either you can add an edge to $$A$$ by keeping these two corrects, or you can remove an edge $$xy$$ form $$A$$ and add some new edges $$vx, yw$$.

I'm using this existential theorem in the following context: take $$T$$ copies of cycles $$C_1, \dots, C_T$$ of size $$n/T$$ and chose $$T$$ such that we can apply the theorem (hence $$T = \theta(n/\log n)$$).

Questions: what combinatorial properties can one expect from a graph $$G$$ produced by this theorem? Note that if one may find some properties by looking at the graphs exactly produced by the theorem, I'm only interested in combinatorial properties at the moment.

I already figured out the following:

1. as $$G$$ is cubic, each cycle as $$T$$ edges going to some other cycles. Because there's $$T-1$$ other cycles, it means that for each cycle, there exist an other cycle s.t. at least two edges going towards the same cycle.
2. By choosing an appropriate $$n$$, we can have $$n/T$$ being odd, forcing the existence of $$p\ge 3$$ cycles $$C'_1, ... C'_p$$. Let $$\tilde{G}$$ bet the graph where each node is on of $$C'_{i}$$ and an edge exist between two nodes if there are two edges between corresponding cycles in $$G$$. $$\tilde{G}$$ is connected (but I think we can't tell if it has cycle or it is a tree.)