ASYMPTOTIC NUMBERS OF GENERAL 4-REGULAR GRAPHS WITH GIVEN CONNECTIVITIES

Let g(n;l1;l2;d;t;q) be the number of general 4-regular graphs on n labelled vertices with l1 + 2l2 loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with flve types of properties to determine the asymptotic behav- ior of g(n;l1;l2;d;t;q) and hence that of g(2n), the total number of general 4-regular graphs where l1, l2, d, t and q = o( p n); re- spectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.