In 1968, Erdös and Lovász conjectured that for every graph G and all integers s, t ≥ 2 such that s + t − 1 = χ(G) > ω(G), there exists a partition (S, T ) of the vertex set of G such that χ(G|S) ≥ s and χ(G|T ) ≥ t. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stabili...

Let Ψ : Z → Z be an affine-linear map, and Ψ̇ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z) of Z. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture MN(s) for some finite s > 1, GreenTao [GT3] verified Dickson’s conjecture for Ψ of complexity s. Let Φ : Z → Z be an...

Kollár’s conjecture states that a complex projective surface S with quotient singularities and with H(S, Q) = Q should be rational if its smooth part S is simply connected. We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of S has at most 3 components over each singular point of S.

We give many examples of surfaces of general type with pg = 0 for which Bloch’s conjecture holds, for all values of K2 6= 9. Our surfaces are equipped with an involution. Let S be a smooth complex projective surface with pg(S) = 0. Bloch’s conjecture states that the Albanese map A0(S)0 → Alb(S) is an isomorphism, where A0(S)0 is the Chow group of 0-cycles of degree 0 on S. It is known for all s...

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

in this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable taylor and weight function approximation. the method is based on weerakoon-fernando method [s. weerakoon, g.i. fernando, a variant of newton's method with third-order convergence, appl. math. lett. 17 (2000) 87-93]. the method is optimal, as it needs three evaluations per ite...

The study of zero-divisors in group rings had become interesting problem since 1940 with the famous zero-divisor conjecture proposed by G.Higman [2]. Since then several researchers [1, 2, 3] have given partial solutions to this conjecture. Till date the problem remains unsolved. Now we introduce the notions of Smarandache zero divisors (S-zero divisors) and Smarandache week zero divisors (S-wea...