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Number of connected 4-regular simple graphs on n vertices with girth at least 4.
20

%I #30 Feb 01 2018 01:05:23

%S 1,0,0,0,0,0,0,0,1,0,2,2,12,31,220,1606,16828,193900,2452818,32670330,

%T 456028474,6636066099,100135577747,1582718912968

%N Number of connected 4-regular simple graphs on n vertices with girth at least 4.

%C The null graph on 0 vertices is vacuously connected and 4-regular; since it is acyclic, it has infinite girth. - _Jason Kimberley_, Jan 29 2011

%D M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), pp. 137-146.

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_girth_ge_4">Connected regular graphs with girth at least 4</a>

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_ge_g_index">Index of sequences counting connected k-regular simple graphs with girth at least g</a>

%H M. Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a>

%Y Contribution from _Jason Kimberley_, Mar 19 2010 and Jan 28 2011: (Start)

%Y 4-regular simple graphs with girth at least 4: this sequence (connected), A185244 (disconnected), A185344 (not necessarily connected).

%Y Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), this sequence (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).

%Y Connected 4-regular simple graphs with girth at least g: A006820 (g=3), this sequence (g=4), A058343 (g=5), A058348 (g=6).

%Y Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5). (End)

%K nonn,nice,more,hard

%O 0,11

%A _N. J. A. Sloane_, Dec 17 2000

%E By running M. Meringer's GENREG at U. Newcastle for 6.25, 107 and 1548 processor days, a(21), a(22), and a(23) were completed by Jason Kimberley on Dec 06 2009, Mar 19 2010, and Nov 02 2011.