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

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of Pv-factorization of Km,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 P4k 1-factorization of Km,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.     

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the Boolean lattice is discussed. By using this correspondence, we show existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

An alternative proof of Painlevé's theorem

In this article we show some aspects of analytical and numerical solution of the n-body problem, which arises from the classical Newtonian model for gravitation attraction. We prove the non-existence of stationary solutions and give an alternative proof for Painlevé's theorem.     

Non-existence of bipartite graphs of diameter at least 4 and defect 2

The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Δ and diameter D. Bipartite graphs of maximum degree Δ, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Δ such that Δ−1 is a prime power.     

An alternative proof of the Aubin–Lions lemma

Using a generalized version of the Weyl–Riesz criterion for compactness of subsets of Lebesgue–Bochner spaces, we present in this short note an alternative proof of a result by J. Simon [4] that extends the classical result by J.P. Aubin and J.L. Lions on compact embeddings in Lebesgue–Bochner spaces to the non-reflexive Banach space case.     

A generalization of Eagon–Reiner’s theorem and a characterization of bi-CMt bipartite and chordal graphs

We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CM_t property of its Alexander dual. Then we characterize bi-CM_t bipartite graphs and bi-CM_t chordal graphs. These are generalizations of recent results due to Herzog and Rahimi.     

An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality

We give a short alternative proof of Berg and Nikolaev’s recent theorem on a characterization of CAT(0)-spaces via the quadrilateral inequality.     

An alternative proof of Pellet’s theorem for matrix polynomials

We propose an alternative proof of Pellet’s theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouché’s theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet’s theorem.     

An abstract characterization of Thompson’s group F

We show that Thompson’s group F is the symmetry group of the ‘generic idempotent’. That is, take the monoidal category freely generated by an object A and an isomorphism A ? A→A; then F is the group of automorphisms of A.     

The critical groups of a family of graphs and elliptic curves over finite fields

Let q be a power of a prime, and E be an elliptic curve defined over  . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions for all k≥1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius map of E over  . This work was partially supported by the NSF, grant DMS-0500557 during the author’s graduate school at the University of California, San Diego, and partially supported by an NSF Postdoctoral Fellowship.     

Graham’s pebbling conjecture on product of complete bipartite graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs G and H, f( G x H) ⩽ f( G) f( H). We show that Graham’s conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are complete bipartite graphs.     

P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n.When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k - 1)m ≤ 2kn, (2) (2k -1)n≤2km, (3) m n ≡ 0 (mod 4k - 1), (4) (4k -1)mn/[2(2k -1)(m n)] is an integer.     

An elementary proof of a distribution relation¶on elliptic curves

We give another elementary proof of a certain identity of elliptic functions arising from the K-theory of elliptic curves and Wildeshaus's generalisation of Zagier's conjectures. This proof consists of a calculation with the q-expansions, and is offered in the hope that its more explicit flavour may be generalised to other situations. Received: 7 December 1999 / Revised version: 3 July 2000     

D-saturated property of the Cayley graphs of semigroups

Let D be a finite graph. A semigroup S is said to be Cayley D-saturated with respect to a subset T of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S,T) isomorphic to D with all vertices in V. The purpose of this paper is to characterize the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S. In particular, the Cayley D-saturated property of a semigroup S with respect to any subsemigroup T is characterized.     

Edge-decompositions of highly connected graphs into paths

We prove that a graph of edge-connectivity at least has an edge-decomposition into paths of length 4 if and only its size is divisible by 4. We also prove that a graph of girth >m and of edge-connectivity at least 8 ^m has an edge-decomposition into paths of length m provided its size is divisible by m, and m is a power of 2.      

On 2-factors with cycles containing specified vertices in a bipartite graph

Let k≥1 be an integer and G=(V ₁,V ₂;E) a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+2. Our result is as follows: If $d(x)+d(y)\geq \lceil\frac{4n+k}{3}\rceil$ for any nonadjacent vertices x∈V ₁ and y∈V ₂, then for any k distinct vertices z ₁,…,z _k , G contains a 2-factor with k+1 cycles C ₁,…,C _k+1 such that z _i ∈V(C _i ) and l(C _i )=4 for each i∈{1,…,k}.     

An elementary proof of Tverberg’s theorem

A new proof is suggested for Tverberg’s familiar theorem saying that an arbitrary set of q: (d + 1)(p − 1) + 1 points in ℝ ^d can be split into p parts such that their convex hulls have a nonempty intersection. Bibliography: 9 titles.