On the critical graph conjecture

Gol'dberg has recently constructed an infinite family of 3-critical graphs of even order. We now prove that if there exists a p(≥4)-critical graph K of odd order such that K has a vertex u of valency 2 and another vertex v ≠ u of valency ≤(p + 2)/2, then there exists a p-critical graph of even order.     

On the number of Morse points for deformations of real hyperplane singularities

A relation for the algebraic number of Morse points for deformations of real hyperplane singularities is obtained. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

On Meyniel's conjecture of the cop number

Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper‐bound O(n/ lnn) due to Chiniforooshan.     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

On the strong perfect graph conjecture and critical graphs

In this paper we obtain some properties of graphs which are critical with respect to perfectness. Some alternate forms of Berge's [2] strong perfect graph conjecture are also given.     

Nagata’s conjecture for a square or nearly-square number of points

Abstract In this paper we prove that a linear system of plane curves of degree d with n points of multiplicity m always has the expected dimension when n is a perfect square. We also prove some results on the dimension of the linear system when n is close to a perfect square. Keywords: Interpolation, Nagata, Linear systems, Plane curves Mathematics Subject Classification (2000): 14J26, 14N05     

On Darboux points of real functions

The nature and relationship of the sets of Darboux points and continuity points of real function are studied.     

On the equivariant Tamagawa number conjecture in tame CM-extensions

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair ${(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}$ naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in ${K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}$ which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer?CStark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.     

On the Equivariant Tamagawa number conjecture for Tate motives

Let L be a finite abelian extension of and let K be any subfield of L. For each integer r with r0 we prove the Equivariant Tamagawa Number Conjecture for the pair (h⁰(Spec(L))(r),[&frac;][Gal(L/K)]). Mathematics Subject Classification (1991) 11G40, 11R23, 11R33, 11R65     

A conjecture regarding the logarithmic coefficients of univalent functions

One considers the class S of functions, regular and univalent in ¦Z¦<1 and normalized by the expansion f(z)=Z + C₂Z² +.... By the logarithmic coefficients of the function f (z) S one means the coefficients of the expansion Earlier, the author had formulated the following conjecture: for any function f(z) S, for each z (0,1) one has the inequality In this paper this conjecture is proved for spiral-shaped functions and for functions from S with real coefficients and under some additional assumptions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 135–143, 1983.