12 research outputs found
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.</jats:p
An efficient algorithm to test forcibly-connectedness of graphical degree sequences
We present an algorithm to test whether a given graphical degree sequence is
forcibly connected or not and prove its correctness. We also outline the
extensions of the algorithm to test whether a given graphical degree sequence
is forcibly -connected or not for every fixed . We show through
experimental evaluations that the algorithm is efficient on average, though its
worst case run time is probably exponential. We also adapt Ruskey et al's
classic algorithm to enumerate zero-free graphical degree sequences of length
and Barnes and Savage's classic algorithm to enumerate graphical partitions
of even integer by incorporating our testing algorithm into theirs and then
obtain some enumerative results about forcibly connected graphical degree
sequences of given length and forcibly connected graphical partitions of
given even integer . Based on these enumerative results we make some
conjectures such as: when is large, (1) almost all zero-free graphical
degree sequences of length are forcibly connected; (2) almost none of the
graphical partitions of even are forcibly connected.Comment: 20 pages, 11 table
An Efficient Algorithm to Test Forcibly-connectedness of Graphical Degree Sequences
We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly -connected or not for every fixed . We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al\u27s classic algorithm to enumerate zero-free graphical degree sequences of length and Barnes and Savage\u27s classic algorithm to enumerate graphical partitions of even integer by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length and forcibly connected graphical partitions of given even integer . Based on these enumerative results we make some conjectures such as: when is large, (1) almost all zero-free graphical degree sequences of length are forcibly connected; (2) almost none of the graphical partitions of even are forcibly connected
Deciding football sequences
An open problem posed by the first author is the complexity to decide whether
a sequence of nonnegative integer numbers can be the final score of a football
tournament. In this paper we propose polynomial time approximate and
exponential time exact algorithms which solve the problem