For a graph G, χ(G) (ω(G)) denote its chromatic (clique) number. A P5 is the chordless path on five vertices, and 4-wheel consisting of cycle four vertices C4 plus an additional vertex adjacent to all C4. In this paper, we show that every (P5, 4-wheel)-free G satisfies χ(G)≤32ω(G). Moreover, bound almost tight. That is, there class graphs L such H∈L χ(H)≥107ω(H). This generalizes/improves sever...

This upper bound is used to calculate several chromatic numbers of strong products. It is shown in particular that for k ≥ 2, χ(C5 2 ×C5 2 ×C2k+1) = 10 + d 5 ke, and for k ≥ 2 and n ≥ 1, χ(C2k+1 2 ×Kn) = k n + d2 e. That the general upper bound cannot be improved for graphs which are not χ–critical is demonstrated by two infinite series of graphs. The paper is concluded with an application to g...

Values of Dirichlet L-functions at s = 1 have always attracted great attention due to their central role in number theory. In particular the prime number theorem for arithmetic progressions relies on the non-vanishing of L(1, χ) for any non-principal character χ (mod q). Moreover improving the existing lower bounds for |L(1, χ)| (the famous Siegel’s bound |L(1, χ)| ≫ǫ q) would imply many import...

Let F be a totally real field and χ an abelian totally odd character of F . In 1988, Gross stated a p-adic analogue of Stark’s conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross’s conjecture when F is a real quadratic field and χ is a narrow ring class ...

Studies have revealed that elevated homocysteine levels can cause damage to motor neurons through multiple neurotoxic mechanisms, thus leading to the pathogenesis of amyotrophic lateral sclerosis (ALS). One way by which homocysteine levels are increased in the body is the consequence of methylenetetrahydrofolate reductase (MTHFR) gene polymorphisms. Therefore, to address this question, we studi...

In this paper, we present appendices employed in the paper ”A geometric level-set formulation of a plasma-sheath interface” by the authors. Appendix A. Local existence of sheath solutions In this appendix, we present a series of a priori estimates for the approximate solutions constructed in Step 0 Step 7 in Section 7 and then give the proof Theorem 7.3. A.1. Basic a priori estimates. In this p...

We investigate the factors influencing effective stress parameter, χ, in unsaturated granular soils using pore-scale numerical modelling. The Shan-Chen multiphase lattice Boltzmann method (LBM) is used for this purpose. A procedure outlined measuring χ LBM data. full hydraulic cycle simulated a soil column and individual contributions of suction surface tension forces to are measured as functio...

It is well known that if G is a multigraph then χ(G) ≥ χ(G) := max{∆(G), Γ(G)}, where χ(G) is the chromatic index of G, χ(G) is the fractional chromatic index of G, ∆(G) is the maximum degree of G, and Γ(G) = max{2|E(G[U ])|/(|U | − 1) : U ⊆ V (G), |U | ≥ 3, |U | is odd}. The conjecture that χ(G) ≤ max{∆(G) + 1, dΓ(G)e} was made independently by Goldberg (1973), Anderson (1977), and Seymour (19...

We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ G , to a new graph parameter Ψβ(G), nested between ω(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter...