Let G be a simple graph of order n, and let ∆(G) and χ′(G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ′(G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ′(G) = ∆(G) + 1 and χ′(G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic critical, then the independence number α(G) ≤ n/2, ...

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s+ t = χ(G)+1, there is a partition (S, T ) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for line graphs of multigraphs.

The Kneser graph K(n, k) is the graph whose vertices are the k-elements subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(2k + 1, k) is an...

OBJECTIVE The objective of this study was to investigate predictor variables at the moment of normal stress-rest myocardial perfusion gated SPECT for indication of a second gated SPECT. MATERIAL AND METHODS A prospective, single center cohort study was conducted. We evaluated 2326 consecutive patients (age 63.6 ± 13 years, 57.3% females) without perfusion defects and with normal left ventricu...

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta funct...

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S, T ) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2. AMS Subject C...

Let χ be a nontrivial multiplicative character of Fpn . We obtain the following results. (1). Let ε > 0 be given. If B = {Pnj=1 xjωj : xj ∈ [Nj + 1, Nj + Hj ] ∩ Z, j = 1, . . . , n} is a box satisfying n Π j=1 Hj > p ( 2 5, then for p > p(ε) we have, denoting χ a nontrivial multiplicative character | X x∈B χ(x)| ¿n p− ε 2 4 |B| unless n is even, χ is principal on a subfield F2 of size pn/2 and ...