Rainbow and orthogonal paths in factorizations of Kn

For n even, a factorization of a complete graph K_n is a partition of the edges into n?1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizations have orthogonal paths. Assisted by a simple edge‐switching algorithm, here we show that for n?8, the rotational factorization of K_n, GK_n has orthogonal paths. We prove that this algorithm finds a rainbow path with at least (2n+1)/3 vertices in any factorization of K_n (in fact, in any proper coloring of K_n). We also give some problems and conjectures about the properties of the algorithm. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 167–176, 2010     

On isomorphic factorizations of circulant graphs

We investigate the conjecture that every circulant graph X admits a k‐isofactorization for every k dividing |E(X)|. We obtain partial results with an emphasis on small values of k. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 406–414, 2006     

On minimal factorizations of rational matrix functions

The theory of factorization of rational matrix functions W() = = C(I-A)^–1B + D, as presented in the book Bart-Gohberg-Kaashoek [1], is extended here to the case where D = W() is not invertible, and applied to factorizations of monic matrix polynomials.     

On some factorizations of two-parameter polynomial matrices

This paper is a logical continuation of the author's discussion about the solution of spectral problems for two-parameter polynomial matrices of general type. Various rank factorization algorithms are suggested, among them the so-called minimal factorization of a singular two-parameter polynomial matrix of degenerate rank into a product of some matrices whose ranks are equal to the rank of the original matrix. Spectral properties of these matrices are studied. The notion of minimal factorization is also extended to one-parameter polynomial and constant matrices. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 94–116 Translated by V. N. Kublanovskaya.     

On the decomposition of Kn into complete m-partite graphs

Graham and Pollak [3] proved that n ?1 is the minimum number of edge-disjoint complete bipartite subgraphs into which the edges of K_n can be decomposed. Using a linear algebraic technique, Tverberg [2] gives a different proof of that result. We apply his technique to show that for “almost all n,” ? (n + m ?3)/(m ?1) ? is the minimum number of edge-disjoint complete m-partite subgraphs in a decomposition of K_n.     

On DRC‐covering of Kn by cycles

This paper considers the cycle covering of complete graphs motivated by the design of survivable WDM networks, where the requests are routed on INF‐networks which are protected independently from each other. The problem can be stated as follows: for a given graph G, find a cycle covering of the edge set of K_n, where V(K_n) = V(G), such that each cycle in the covering satisfies the disjoint routing constraint (DRC7rpar;, relatively to G, which can be stated as follows: to each edge of K_n we associate in G a path and all the paths associated to the edges of a cycle of the covering must be vertex disjoint. Here we consider the case where G = C_n, a ring of size n and we want to minimize the number of cycles in the covering. We give optimal solutions for the problem as well as for variations of the problem, namely, its directed version and the case when the cycle length is fixed to 4. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 100–112, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10040     

Edge‐decompositions of Kn,n into isomorphic copies of a given tree

We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C₆‐free graphs with Ω(n^4/3 edges, C₁₀‐free graphs with Ω(n^6/5) edges, and Θ(7,7,7)‐free graphs with Ω(n^8/7) edges. Each of these bounds is sharp in order of magnitude. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 1–10, 2005     

On systolic array methods for band matrix factorizations

A systolic array for band matrixQR factorization is presented and its connection with various other arrays explained. It is shown how the computations should be performed on banded rectangular matrices which appear in several numerical problems.     

On the number of factorizations of a full cycle

We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending recent work of Biane. We deduce from our result a remarkable formula of Poulalhon and Schaeffer that was previously derived through an intricate combinatorial argument.     

On irreducible factorizations of rational matrices and their applications

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 117–156. This work was carried out during our visit to Sweden under the financial support of the Chalmer University of Technology in Góterborg and the Institute of Information Processing of the University of Umeă. Translated by V. N. Kublanovskaya.     

Simulated factorizations II

Summary A factorization of a finite abelian group is said to be simulated if it is obtained from a factorization into a direct product of subgroups by changing at mostk elements in each subgroup. The question has been asked as to which values ofk imply that in fact at least one subgroup must be left unaltered. This has been shown to be true fork = 1 but to be false, in general, fork = p – 1, wherep is the least prime dividing the order ofG. In this paper it is shown to be true fork = p – 2.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

Orthogonal factorizations of graphs

Let G be a graph with vertex set V(G) and edge set E(G). Let k₁, k₂,…,k_m be positive integers. It is proved in this study that every [0,k₁+…+k_m?m+1]‐graph G has a [0, k_i]₁^m‐factorization orthogonal to any given subgraph H with m edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 267–276, 2002     

Isomorphic factorizations of trees

A tree is even if its edges can be colored in two colors so that the monochromatic subgraphs are isomorphic. All even trees of maximum degree 3 in which no two vertices of degrees 1 or 3 are adjacent are determined. It is also shown that, for every n, there are only finitely many trees of maximum degree 3 and with n vertices of degree 3 that are not even. © 1995 John Wiley & Sons, Inc.     

关于超魔图SPE(K_n,f)的几个结果

在引入裂边拓展图概念的基础上 ,给出了图 SPE( K4 ,f )是超魔图的充要条件 ,得到了图SPE( Kn,f ) ( n =3 ,4,5 )是超魔图的几个结果 ,指出了其一般形式下标号集的构造 ,并将已有的结论作了进一步的推广 .     

On the asymptotic behaviour of the number of distinct factorizations into irreducibles

For an integral domainR and a non-zero non-unitaεR we consider the number of distinct factorizations ofa ⁿ into irreducible elements ofR for largen. Precise results are obtained for Krull domains and certain noetherian domains. In fact, we prove results valid for certain classes of monoids which then apply to the above-mentioned classes of domains.