Rational matrix groups of a special type

Let G be a finite group having a faithful irreducible character χ for which χ(1) is prime to ¦G¦/χ(1). Let n=[$\text{Q}$(χ):$\text{Q}$]χ(1), and assume that the factors are not both even. Then G can be embedded in GL_n($\text{Q}$) in such a way that its normalizer therein splits over its centralizer.     

Majorations explicites de |L(1,χ)| (quatrième partie)Explicit upper bounds on |L(1,χ)| (part four)

We prove that for any even primitive Dirichlet character χ of odd conductor q_χ>1 we have $\text{|(1?}\text{χ(2)}\text{2}\text{)L(1,χ)|?}\text{1}\text{4}\text{(}\text{log}\text{q}{}_{\mathrm{\chi }}\text{+κ),}$ where κ:=2+γ?log(π/4)=2.81878…. To cite this article: S.R. Louboutin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 625–628.     

On the mean square average of special values of L-functions

Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan?s and Euler?s totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r?1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r?3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.     

Locally restricted colorings

We consider the following problem: given suitable integers χ and p, what is the smallest value ρ such that, for any graph G with chromatic number χ and any vertex coloring of G with at most χ+p colors, there is a vertex v such that at least χ different colors occur within distance ρ of v? Let ρ(χ,p) be this value; we show in particular that ρ(χ,p)?⌈p/2⌉+1 for all χ,p. We give the exact value of ρ when p=0 or χ?3, and (χ,p)=(4,1) or (4,2).     

On the chromatic index of multigraphs without large triangles

Let M be a multigraph. Vizing (Kibernetika (Kiev)1 (1965), 29–39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where $\text{1}\text{2}\text{(μ(M) + 1) < s}$ then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory4 (1980), 265–268) had conjectured that (C) if G is a simple graph that does not induce K_1,3 or K₅?e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math.45 (1983), 277–285) proved that (C′) implies (C); thus Javedekar's conjecture is true.     

On regular bipartite-preserving supergraphs

For a graphG with chromatic numberχ(G) ? 2 and maximum degree Δ(G), there exists anr-regular graphH, for everyr ? Δ(G), such thatG is an induced subgraph ofH and_χ(H) =_χ (G). In the case whereG is bipartite, the minimum order of such a graphH is determined.     

Relative colorings of graphs

In this paper, the notion of relative chromatic number χ(G, H) for a pair of graphs G, H, with H a full subgraph of G, is formulated; namely, χ(G, H) is the minimum number of new colors needed to extend any coloring of H to a coloring of G. It is shown that the four color conjecture (4CC) is equivalent to the conjecture (R4CC) that χ(G, H) ≤ 4 for any (possibly empty) full subgraph H of a planar graph G and also to the conjecture (CR3CC) that χ(G, H) ≤ 3 if H is a connected and nonempty full subgraph of planar G. Finally, relative coloring theorems on surfaces other than the plane or sphere are proved.     

A short proof for the nonvanishing of a character sum

Let χ be an odd residue class character with conductor f (a prime power) and order 2^h. It is shown that the nonvanishing of the sum χ(1) + 2χ(2) + … + fχ(f) follows directly from properties of the minimum polynomial of a 2^hth root of unity.     

Inequalities and identities for generalized matrix functions

Let χ be a character on the symmetric group S_n, and let A = (a_ij) be an n-by-n matrix. The function d_χ(A) = Σ_σ?Snχ(σ)Πⁿ_{t = 1}a_tσ(t) is called a generalized matrix function. If χ is an irreducible character, then d_χ is called an immanent. For example, if χ is the alternating character, then d_χ is the determinant, and if χ ≡ 1, then d_χ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality $\text{(1/χ(}\text{id}\text{))d}{}_{\mathrm{\chi }}\text{(A) ?}\text{per}\text{A}$ holds for a variety of characters χ including the irreducible ones corresponding to the partitions (n ? 1,1) and (n ? 2,1,1) of n. The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.     

Values of twisted Artin L-functions

This note gives a simple proof that certain values of Artin’s L-function, for a representation ρ with character χ _ρ , are stable under twisting by an even Dirichlet character χ, up to the dim(ρ)th power of the Gauss sum τ(χ) and an element generated over \({\mathbb{Q}}\) by the values of χ and χ _ρ . This extends a result due to J. Coates and S. Lichtenbaum.     

The chromatic polynomial of an unlabeled graph

We investigate the chromatic polynomial χ(G, λ) of an unlabeled graph G. It is shown that $\text{χ(G, λ) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{χ(g, π, λ)}$, where g is any labeled version of G, A(g) is the automorphism group of g and χ(g, π, λ) is the chromatic polynomial for colorings of g fixed by π. The above expression shows that χ(G, λ) is a rational polynomial of degree n = |V(G)| with leading coefficient $\text{1}\text{|A(g)|}$. Though χ(G, λ) does not satisfy chromatic reduction, each polynomial χ(g, π, λ) does, thus yielding a simple method for computing χ(G, λ). We also show that the number N(G) of acyclic orientations of G is related to the argument λ = ?1 by the formula $\text{N(G) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{(?1)}{}^{\mathrm{s(\pi )}}\text{χ(g, π, ?1)}$, where s(π) is the number of cycles of π. This information is used to derive Robinson's (“Combinatorial Mathematics V” (Proc. 5th Austral. Conf. 1976), Lecture Notes in Math. Vol. 622, pp. 28–43, Springer-Verlag, New York/Berlin, 1977) cycle index sum equations for counting unlabeled acyclic digraphs.     

An adelic Hankel summation formula

In this paper, the convergence of the Euler product of the Hecke zeta-function ζ(s,χ) is proved on the line R(s)=1 with s≠1. A certain functional identity between ζ(s,χ) and ζ(2−s,χ) is found. An analogue of Tate's adelic Poisson summation is obtained for the global Hankel transformation, which is constructed in Li (2010) [7].     

Hook-free colorings and a problem of Hanson

Hanson posed the following problem: What is the minimum numberχ(n) of colors needed to color all subsets of ann-set such that there is no monochromatic tripleA, B, C withA ∪B=C? It is known thatχ(n)≦[(n+1)/2], while Erd?s and Shelah provedχ(n)≧[(n+1)/4]. Their proof suggests the following notion: LetC be any finite plane point-configuration. The hook-free coloring numberχ(C) is the smallest number of colors needed forC such that no monochromatic hooks arise, i.e. if (c _x ,c _y ) are the coordinates of pointc∈C, then there are no 3 distinct pointsa, b, c∈C witha _x =b _x <c _x ,b _y =c _y <a _y . In this paperχ(R _m,n ) is determined exactly for anm×n-rectangle, and asymptotically for the triangular staircase. As a corollary one obtainsχ(n)≧0.293n.     

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ_l(G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l≥3), χ_l(G)≤3.     

Second moments of Dirichlet L-functions weighted by Kloosterman sums

For the general modulo q ? 3 and a general multiplicative character χ modulo q, the upper bound estimate of |S(m, n, 1, χ, q)| is a very complex and difficult problem. In most cases, the Weil type bound for |S(m, n, 1, χ, q)| is valid, but there are some counterexamples. Although the value distribution of |S(m, n, 1, χ, q)| is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for k-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet L-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y.Yi, X.He: On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.     

一类连通无三角形图线图的共色数的下界

Erd(o)s,Gimbel and Straight (1990) conjectured that if ω(G)＜5 and z(G)＞3,then z(G)≥χ(G)-2. But by using the concept of edge cochromatic number it is proved that if G is the line graph of a connected triangle-free graph with ω(G)＜5 and G≠K4, then z(G)≥χ(G)-2.     

The fractional chromatic number, the Hall ratio, and the lexicographic product

Let χ_f denote the fractional chromatic number and ρ the Hall ratio, and let the lexicographic product of G and H be denoted GlexH. Main results: (i) ρ(GlexH)≤χ_f(G)ρ(H); (ii) if ρ(G)=χ_f(G) then ρ(GlexH)=ρ(G)ρ(H) for all H; (iii) χ_f−ρ is unbounded. In addition, the question of how big χ_f/ρ can be is discussed.     

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue characters

Let χ be a primitive quadratic residue class character with conductor f. It is shown that the generalized Bernoulli polynomials B_χⁿ(x) and the related polynomials B_χⁿ(x) − B_χⁿ(0), divided by obvious factors, are irreducible over the rationals under certain conditions on f and n.     

Inequalities for the zeros of dirichlet L-functions

Let χ denote a primitive, Dirichlet character to the modulus q>i and let L(s,χ) be the corresponding Dirichlet L-series defined by L(s,χ) = ∑χ(n)n^?s,s = σ+it, for σ>0. It is of interest to know where the zeros of L(s,χ) are located, since the location of these zeros would yield important results in number theory. In this paper, we show that the spectrum of each member of a certain class of Hermitian matrices leads to an explicit zero-free region for L(s,χ).     

Chromatic number and complete graph substructures for degree sequences

Given a graphic degree sequence D, let χ(D) (respectively ω(D), h(D), and H(D)) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is D. It is proved that χ(D)≤h(D). Moreover, it is shown that a subdivision of a clique of order χ(D) exists where each edge is subdivided at most once and the set of all subdivided edges forms a collection of disjoint stars. This bound is an analogue of the Hajós Conjecture for degree sequences and, in particular, settles a conjecture of Neil Robertson that degree sequences satisfy the bound χ(D) ≤ H(D) (which is related to the Hadwiger Conjecture). It is also proved that χ(D) ≤ 6/5 ω(D)+ 3/5 and that χ(D) ≤ 4/5 ω(D) + 1/5 Δ(D)+1, where Δ(D) denotes the maximum degree in D. The latter inequality is related to a conjecture of Bruce Reed bounding the chromatic number by a convex combination of the clique number and the maximum degree. All derived inequalities are best possible