Let G be a graph, G its complement, L(G) its line graph, and χ(G) its chromatic number. Then we have the following Theorem Let G be a graph with n vertices. (i) If G is triangle free, then n− 4 ≤ χ ( L(G) ) ≤ n− 2 (ii) If G is planar and every triangle bounds a disk, then n− 3 ≤ χ ( L(G) ) ≤ n− 2

We say that two elliptic curves E1, E2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χ of K, there is an isomorphism Seln(E χ 1 /K) ∼= Seln(E 2 /K) between the nSelmer groups of the quadratic twists E 1 , E χ 2 . We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenou...

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

Using telephone survey data from 1111 retired older adults (≥65 years; 634 male, 477 female), we tested the hypothesis that exposure to shift work might result in increased self-reported diabetes. Five shift work exposure bins were considered: 0 years, 1-7 years, 8-14 years, 15-20 years, and >20 years. Shift work exposed groups showed an increased proportion of self-reported diabetes (χ(2) = 22...

The prospects for detecting trilepton events (ℓ = e or µ) from chargino-neutralino (χ ± 1 χ 0 2) associated production are investigated for the upgraded Fermilab Tevatron Collider in the context of the minimal supergravity model (mSUGRA). In some regions of parameter space, χ ± 1 and χ 0 2 decay dominantly into final states with τ leptons and the contributions from τ −leptonic decays enhance th...

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ n+ω+1−α 2 . Moreover, χ(G) ≤ n+ω−α 2 , if either ω + α = n + 1 and G is not a split graph or α+ω = n−1 and G contains no induced Kω+3−C5.

A triangulation of a closed surface is said to be d-covered if at least one vertex in each edge has degree d. In [7] the authors have shown that: (a) if a closed surface of non-positive Euler characteristic χ admits a dcovered triangulation then d ≤ 2 ⌊ √ 49−24χ 2 ⌋, and (b) the upper bound for d is attained when: (i)χ = 0 and for (ii) d = 2(n − 1), whenever n = 7+ √ 49−24χ 2 is a positive inte...

For the chromatic index χ′(G) of a (multi)graph G, there are two trivial lower bounds, namely the maximum degree ∆(G) and the density W(G) = maxH⊆G,|V (H)|≥2 d|E(H)|/ b|V (H)|/2ce. A famous conjectures due to Goldberg [3] and Seymour [10] says that every graph G satisfies χ′(G) ≤ max{∆(G) + 1,W(G)}. This means that χ′(G) = W(G) for every graph G with χ′(G) ≥ ∆(G) + 2. The considered class of gr...

1. L(s, χ) for even Dirichlet characters 2. L(s, χ) for odd Dirichlet characters 3. Dedekind zeta function ζ o (s) for Gaussian integers o 4. Grossencharacter L-functions L-functions for Gaussian integers We try to imitate the argument used by Riemann for proving the analytic continuation of ζ(s) and its functional equation π −s/2 Γ(s 2) ζ(s) = π −(1−s)/2 Γ(1−s 2) ζ(1 − s) from the integral rep...