The spectrum of path factorization of bipartite multigraphs

LetλK_(m,n)be a bipartite multigraph with two partite sets having m and n vertices, respectively.A P_v-factorization ofλK_(m,n)is a set of edge-disjoint P_v-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P_v-factorization ofλK_(m,n).When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P_(4k 1)-factorization ofλK_(m,n)are(1)2km≤(2k 1)n,(2)2kn≤(2k 1)m,(3)m n≡0(mod 4k 1),(4)λ(4k 1)mn/[4k(m n)]is an integer.     

On the non-existence of pair covering designs with at least as many points as blocks

We establish new lower bounds on the pair covering number C _λ (υ,k) for infinitely many values of υ, k and λ, including infinitely many values of υ and k for λ=1. Here, C _λ (υ,k) denotes the minimum number of k-subsets of a υ-set of points such that each pair of points occurs in at least λ of the k-subsets. We use these results to prove simple numerical conditions which are both necessary and sufficient for the existence of (K _k − e)-designs with more points than blocks.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.     

P4k?1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k−1-factorization of λK _m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

On the Iwasawa invariants of totally real number fields

We shall show two sufficient conditions under which the Iwasawa invariants λ _k and μ _k of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK _n be the composite ofk and thel ⁿ-th cyclotomic extension of the fieldQ of rational numbers. LetC _n be the factor group of thel-class group ofK _n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ₁ be an idempotent of the group ringZ _l[Gal(K ₁/k)] defined in the below. We shall prove λ _k = μ _k =0 if there is a natural numbern such that ε₁ C _n vanishes, under additional conditions concerning ramifications inK _n/k.     

Almost Resolvable Maximum Packings of Complete Graphs with 4-Cycles

If the complete graph K _n has vertex set X, a maximum packing of K _n with 4-cycles, (X, C, L), is an edge-disjoint decomposition of K _n into a collection C of 4-cycles so that the unused edges (the set L) is as small a set as possible. Maximum packings of K _n with 4-cycles were shown to exist by Sch?nheim and Bialostocki (Can. Math. Bull. 18:703–708, 1975). An almost parallel class of a maximum packing (X, C, L) of K _n with 4-cycles is a largest possible collection of vertex disjoint 4-cycles (so with ?/4?{\lfloor/4\rfloor} 4-cycles in it). In this paper, for all orders n, except 9, which does not exist, and possibly 23, 41 and 57, we exhibit a maximum packing of K _n with 4-cycles so that the 4-cycles in the packing are resolvable into almost parallel classes, with any remaining 4-cycles being vertex disjoint. [Note: The three missing orders have now been found, and appear in Billington et al. (to appear).]     

On the replications of certain multiplicative designs

Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA ^tA=diag (k ₁-λ₁,…,k _n-λ_n),k _j>λ_j>0,λ₁=…=λ_e,λ_e+1=…=λ_n are obtained which extend previous results for such matrices.     

Sugli zeri delle funzioni zeta di Dedekind

Riassunto Scopo di questo lavoro è dare una formula asintotica per il numero degli zeri di ReF _K(λ+it) e di ImF _K(λ+it), dove eζ _K(8) è la funzione zeta di Dedekind associata al campo numericoK, con 0<t<T e λ numero reale fissato tale che 1−1/n<λ<1 doven è il grado diK.

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Summary The aim of this paper is to give an asymptotic formula for the number of zeros of ReF _K(λ+it) and ImF _K(λ+it), where andζ _K(8) is the Dedekind zeta function for a number fieldK, with 0<t<T and λ fixed real number such that 1−1/n<λ<1, wheren is the degree ofK.

     

The generalized almost resolvable cycle system problem

Let (X, C) be a k-cycle system of order n, with vertex set X (of cardinality n) and collection of k-cycles C. Suppose n=kq+r where r<k. An almost parallel class of C is a collection of q=(n−r)/k pairwise vertex-disjoint k-cycles of C. Each almost parallel class thus will miss r of the n vertices in X. The k-cycle system (X,C) is said to be almost resolvable if C can be partitioned into almost parallel classes such that the remaining k-cycles are vertex disjoint. (These remaining k-cycles are referred to as a short parallel class.)     

The Decomposition Dimension of Graphs

 For an ordered k-decomposition ? = {G ₁, G ₂,…,G _k } of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G ₁), d(e, G ₂),…,d(e, G _k )), where d(e, G _i ) is the distance from e to G _i . A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(K _n ) ≤⌊(2n + 5)/3⌋ for n≥ 3. Received: June 17, 1998 Final version received: August 10, 1999     

On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

 A t(v,k,λ) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly λ blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(q ⁿ +q ⁿ⁻¹ +⋯+q+1,q ²+q+1, q ⁿ⁻² + q ⁿ⁻³ +⋯+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,λ) design is said to be resolvable if the blocks can be partitioned as ℱ={R ₁,R ₂,…,R _s }, where s=λ(v−1)/(k−1) and each R _i consists of v/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length v on the points and ℱ^σ=ℱ, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=〈σ〉 where σ is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group. Received: November 10, 1999 Final version received: September 18, 2000 Acknowledgments. The author would like to thank A. Munemasa for his assistance in writing computer programs on constructing projective spaces and searching for partial spreads. Moreover, she's thankful to T. Hishida and M.␣Jimbo for helpful discussions and for verifying the results of this paper. Present address: Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City 1108, Philippines. e-mail: jumela@mathsci.math.admu.edu.ph     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

Minimum Resolvable Coverings of K v with Copies of K 4 ? e

Suppose K _v is the complete undirected graph with v vertices and K ₄ − e is the graph obtained from a complete graph K ₄ by removing one edge. Let (K ₄ − e)-MRC(v) denote a resolvable covering of K _v with copies of K ₄ − e with the minimum possible number n(v, K ₄ − e) of parallel classes. It is readily verified that n(v, K₄-e) 3 é2(v-1)/5 ù{n(v, K_4-e) \geq \lceil 2(v-1)/5 \rceil} . In this article, it is proved that there exists a (K ₄ − e)-MRC(v) with é2(v-1)/5 ù{\lceil 2(v-1)/5 \rceil} parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 108, 172, 228, 292, 296, 308, 412. In addition, the known results on the existence of maximum resolvable (K ₄ − e)-packings are also improved.     

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs

Let K _m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P _v -factorization of K _m,n is a set of edge-disjoint P _v -factors of K _m,n which partition the set of edges of K _m,n . When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of P _v -factorization of K _m,n . When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio’s conjecture is true when v = 4k − 1. In this paper we shall show that Ushio Conjecture is true when v = 4k − 1, and then Ushio’s conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P _4k+1-factorization of K _m,n is (i) 2km≤(2k+1)n, (ii) 2kn≤(2k+1)m, (iii) m+n≡0 (mod 4k+1), (iv) (4k+1)mn/[4k(m+n)] is an integer.     

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Constants of simultaneous extension of continuous functions

It is shown that ifB is the unit ball of a non-separable Hilbert space with its weak topology, then for every number λ≧1, there exists a spaceK _λ containingB, such that the constant of simultaneous extension fromC(B) toC(K _λ) is exactly λ. This gives a negative answer to the question whether the constants of simultaneous extension ought to be odd integers, as was suggested by examples of Corson-Lindenstrauss and Corson-Pelczynski. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.