The existence of a perfect -factorization of the complete graph with nodes, namely, , for arbitrary even number , is a 40-year-old open problem in graph theory. So far, two infinite families of perfect -factorizations have been shown to exist, namely, the factorizations of and , where is an arbitrary prime number . It was shown in previous work that finding a perfect -factorization of is relate...

We show that a necessary and sufficient condition for the existence of a Kp ,2q factorization of the symmetric complete tripartite digraph K~1,n2,n3 is (i) ni = n2 = n3 == 0 (mod p) for p = q, (ii) ni = n2 = n3 == 0 (mod dp'q'(p' + 2q')) for p =Iq and p' odd, (iii) ni = n2 = n3 == 0 (mod dp'q'(p' + 2q')/2) for p =Iq and p' even, where d = (p, q), p' = p/d, q' = q/d.

A hypergraph H on n nodes is complete 3-uniform if the hyperedges are precisely the set of all node triples. A one-factor of H is a set of hyperedges that cover each node exactly once. According to Baranyai's proof, the hyperedges of H can be partitioned into one-factors iff n ≡ 0 (mod 3). Though the proof is constructive, it leads to an O(2)-time algorithm. We investigate a method for computin...

Recently codes have been developed for computing the Cholesky factorization with complete pivoting of a symmetric positive semidefinite matrix for the serial LAPACK library. In the parallel ScaLAPACK library there are only routines for the unpivoted factorization in the positive definite case and no algorithms use complete pivoting. We aim to assess the feasibility of complete pivoting in ScaLA...

We show that a necessary and sufficient condition for the existence of a Kp,q factorization of the symmetric complete bipartite multi-digraph )"K:n n is (i) m = n == 0 (mod p) for p = q and (ii) m = n == a (mod d(p' + q')p'q'/e) for p =f. q, where d = (p,q), p' = p/d, q' = q/d, e = ()..,p'q').

We show that necessary and sufficient conditions for the existence of a semi-evenly partite star factorization of the symmetric complete tripartite digraph K~I,n2,n3 are (i) k is even, k 2 4 and (ii) nl = n2 = n3 == 0 (mod k(k -1)/3) for k == 0 (mod 6) and nl = n2 = n3 == 0 (mod k(k 1)) for k == 2,4 (mod 6).

Incomplete factorization is one of the most effective general-purpose preconditioning methods for Krylov subspace solvers for large sparse systems of linear equations. Two techniques for enhancing the robustness and performance of incomplete LU factorization for sparse unsymmetric systems are described. A block incomplete factorization algorithm based on the Crout variation of LU factorization ...

F11DNF Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. 1 Purpose F11DNF computes an incomplete LU factorization of a complex sparse non-Hermitian matrix, represented in coordinate storage format. This factorization may be used as a preconditioner in combination w...