Parallel enumeration of degree sequences of simple graphs II

Abstract

In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (Gz(n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is Gz(30) = 5 876 236 938 019 300 and on n = 31 vertices is Gz(31) = 22 974 847 474 172 374. Due to Corollary 21 in [52] these results give the number of degree sequences of simple graphs on 30 and 31 vertices.

Recognition of split-graphic sequences

“…Using the algorithm Erdős-Gallai-Linear [48] or algorithm Havel-Hakimi-Linear [45] we can decide in worst case in O(n) time whether π is graphic.…”

“…This proof is an improved version of the proof of linearity of EGL in [48] and was published in 2012 [45] We exploit that s is monotone and determine the capacity estimations c k = min(jb, s k ) appearing in (1) in constant time. The base of the quick computation is again the sequence of the weight points w(σ) = (w 1 , .…”

“…The main results of the paper are that using different definitions of split graphs [4,5,7,8,9,10,21,22,23,26,28,30,67,82,86,87,88] we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular [17,45] and graphic [43,45,48] sequences. …”

“…See [45] This corollary allows the testing of (a, b, n)-regular sequences in worst case in O(n) time using algorithm Erdős-Gallai-Linear [48] or algorithm HavelHakimi-Chungphaisan [45].…”

“…The following sources contain results on the enumeration of graphic and b-graphic sequences [35,45,46,47,48,49,74].…”