Basically, everything that a regular graph is known to contain will also be present in this broad class of graphs (by simple inclusion).

This is enough to establish some results like Thomassen's, that is, containing a sub-graph of minimum degree >= \delta and "girth" >= g (where you need the degree of the regular graph to be large enough in terms of \delta and g).

*girth of a graph is the length of the smallest cycle (a triangle is a cycle of length 3).