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））.     

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.     

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.     

Ramsey functions involving with large

For fixed integers m,k2, it is shown that the k-color Ramsey number r_k(K_m,n) and the bipartite Ramsey number b_k(m,n) are both asymptotically equal to k^mn as n→∞, and that for any graph H on m vertices, the two-color Ramsey number is at most (1+o(1))n^m+1/(logn)^m-1. Moreover, the order of magnitude of is proved to be n^m+1/(logn)^m if H≠K_m as n→∞.     

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.     

On chromatic uniqueness of two infinite families of graphs

In this paper, it is proven that for each k ≥ 2, m ≥ 2, the graph Θ_k(m,…,m), which consists of k disjoint paths of length m with same ends is chromatically unique, and that for each m, n, 2 ≤ m ≤ n, the complete bipartite graph K_m,n is chromatically unique. © 1993 John Wiley & Sons, Inc.     

K 1,p k-factorization of complete bipartite graphs

Abstract. In this paper, it is shown that a sufficient condition for the existence of a     

Complexes of graph homomorphisms

Hom(G, H) is a polyhedral complex defined for any two undirected graphsG andH. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom(K _m, K_n) is homotopy equivalent to a wedge of (n−m)-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graphG, and integersm≥2 andk≥−1, we have ϖ ₁ ^k (Hom(K _m, G))≠0, thenχ(G)≥k+m; here ℤ₂-action is induced by the swapping of two vertices inK _m, and ϖ₁ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom(G, H) induces a homotopy equivalence. It then follows that Hom(F, K _n) is homotopy equivalent to a direct product of (n−2)-dimensional spheres, while Hom(F, K _n) is homotopy equivalent to a wedge of spheres, whereF is an arbitrary forest andF is its complement. The second author acknowledges support by the University of Washington, Seattle, the Swiss National Science Foundation Grant PP002-102738/1, the University of Bern, and the Royal Institute of Technology, Stockholm.     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if m_a = m_K*d_a*m_K{\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaK\subseteq SU(n)} , for K = SO(n), then m_a^k{\mu _{a}^{k}} is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that m_a^n-1{\mu _{a}^{n-1}} is singular to the Haar measure.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

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     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

Graphic Sequences with a Realization Containing a Union of Cliques

An integer sequence π is said to be graphic if it is the degree sequence of some simple graph G. In this case we say that G is a realization of π. Given a graph H, and a graphic sequence π we say that π is potentially H-graphic if there is some realization of π that contains H as a subgraph. We define σ(H,n) to be the minimum even integer such that every graphic sequence with sum at least σ(H,n) is potentially H-graphic. In this paper, we determine σ(H,n) for the graph H = K_m1∪ K_m2∪...∪ K_mk when n is a sufficiently large integer. This is accomplished by determining σ(K_j + kK₂,n) where j and k are arbitrary positive integers, and considering the case where j = m − 2k and m = ∑ m_i.     

2-Colored triangles in edge-colored complete graphs, II

The largest number n = n(k) for which there exists a k-coloring of the edges of k_n with every triangle 2-colored is found to be n(k) = 2^r5^m, where k = 2m + r and r = 0 or 1, and all such colorings are given. We also prove the best possible result that a k-colored K_p satisfying 1 < k < 1 + √p contains at most k − 2 vertices not in a bichromatic triangle.     

On exactlym times covers

In this paper it is shown that if every integer is covered bya ₁+n ₁ℤ,…,a _k +n _k ℤ exactlym times then for eachn=1,…,m there exist at least ( _n ^m ) subsetsI of {1,…k} such that ∑ _{i ∈ I} 1/n _i equalsn. The bound ( _n ^m ) is best possible. Research supported by the National Nature Science Foundation of P.R. of China.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K₁→K₂→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?.     

Prime T-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k ₀ be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x ₁,…, x _n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x ₁,…, x _n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x ₁,…, x _n ]] = 2m ? 1, if n ≥ 2, K has finite exponent over k ₀, and [k ₀: k] < ∞; (3) t-dim R[[x ₁,…, x _n ]] = 2m, otherwise.     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

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.