Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

On weight distributions of perfect colorings and completely regular codes

A vertex coloring of a graph is called “perfect” if for any two colors a and b, the number of the color-b neighbors of a color-a vertex x does not depend on the choice of x, that is, depends only on a and b (the corresponding partition of the vertex set is known as “equitable”). A set of vertices is called “completely regular” if the coloring according to the distance from this set is perfect. By the “weight distribution” of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring.     

Equitable Partitions of Flat Association Schemes

Let (X, G) be an association scheme. We say that (X, G) is flat if it is homogeneous and if any two distinct points have at most one common g-neighbor for each g ∈ G. In this paper we prove that any nondiscrete equitable partition of (X, G) has at most one singleton if (X, G) is flat, and {X} is the unique equitable partition without any singleton if (X, G) is flat and |X| is a prime. This work was supported for two years by Korea Research Foundation Grant (KRF-2006-003-C00010) and Pusan National University Research Grant. Received: January 31, 2007. Final version received: Novmeber 14, 2007.     

Izergin-Korepin Determinant at a Third Root of Unity

We consider the partition function of the inhomogeneous six-vertex model defined on an n×n square lattice. This function depends on 2n spectral parameters x_i and y_i attached to the respective horizontal and vertical lines. In the case of the domain-wall boundary conditions, it is given by the Izergin-Korepin determinant. For q being an Nth root of unity, the partition function satisfies a special linear functional equation. This equation is particularly simple and useful when the crossing parameter is η = 2π/3, i.e., N = 3. It is well known, for example, that the partition function is symmetric in both the x and the y variables. Using the abovementioned equation, we find that in the case of η = 2π/3, it is symmetric in the union {x} ∪ {y}! In addition, this equation can be used to solve some of the problems related to enumerating alternating-sign matrices. In particular, we reproduce the refined alternating-sign matrix enumeration discovered by Mills, Robbins, and Rumsey and proved by Zeilberger, and we obtain formulas for the doubly refined enumeration of these matrices. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 65–76, January, 2006.     

Some classes of Chebyshev systems

Let L be a linear real space of solutions of a linear homogeneous ordinary differential equation with constant coefficients. This space is a Chebyshev space on any interval of length at most π/ω, where ω is maximal among the imaginary parts of the roots of characteristic polynomial. It turns out that this estimate for the length of the interval is exact in a certain sense. Bibliography: 2 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 35–38.     

Pinching, Pontrjagin classes, and negatively curved vector bundles

We prove several finiteness results for the class ℳ_a,b,π,n of n-manifolds that have fundamental groups isomorphic to π and that can be given complete Riemannian metrics of sectional curvatures within {a,b} where a≤b<0. In particular, if M is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class ℳ_a,b,π1(M),n are total spaces of vector bundles over M. Furthermore, given a word-hyperbolic group π and an integer n there exists a positive ε=ε(n,π) such that the tangent bundle of any manifold in the class ℳ_{-1-ε, -1, π, n} has zero rational Pontrjagin classes. Oblatum 27-IX-1999 & 17-XI-2000?Published online: 5 March 2001     

Ascents of size less than <Emphasis Type="Italic">d</Emphasis> in compositions

A composition of a positive integer n is a finite sequence π₁π₂...π _m of positive integers such that π₁+...+π _m = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if π _i+1 ≥ π _i +d (respectively, π _i < π _i+1 < π _i + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n.     

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.     

The symmetry of the partition function of some square ice models

We consider the partition function Z(N; x ₁ , …, x_N, y ₁ , …, y_N) of the square ice model with domain wall boundary conditions. We give a simple proof that Z is symmetric with respect to all its variables when the global parameter a of the model is set to the special value a = e^iπ/3 . Our proof does not use any determinant interpretation of Z and can be adapted to other situations (e.g., to some symmetric ice models).     

若干联图的均匀全染色

图$G(V,E)$的全色数 $\chi_{t}(G)$就是将$V\bigcup E$分成彼此不相交的全独立分割集的最小个数。 如果任何两个$V\bigcup E$的全独立分割集的元素数目相差不超过1，那么 $V \bigcup E$的全独立分割集的最小个数就称为图$G$的均匀全色数，记为$\chi_{et}(G)$。 在本文中我们给出了当 $m \geq n \geq 3$ 时 $W_m\bigvee K_n$，$F_m \bigvee K_n$及$S_m \bigvee K_n$ 的均匀全色数.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

Approximate values of the multiplicities in the arm of the cocharacter sequence of <Emphasis Type="Italic">M</Emphasis><Subscript>2,1</Subscript>

In this paper we study the cocharacter sequence of M _2,1 and obtain estimates for the multiplicities of the irreducible S _n -characters χ^λ, where λ is any partition with at most 5 parts. The author’s work was supported by both the Faculty Research Council of De Paul University and the National Security Agency, under Grant MDA904-500270. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.     

Soap films bounded by non-closed curves

This paper proves (i) every “geometrically knotted” non-closed curve bounds a soap-film, (ii) any non-closed curve bounding a soap-film must have total curvature greater than 2π, and (iii) for every k > 2π, there is a geometrically knotted non-closed curve with total curvature k.     

Colourings of the cartesian product of graphs and multiplicative Sidon sets

Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic number χ(G,F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free colourings of the cartesian product of graphs.     

Collars and partitions of hyperbolic cone-surfaces

For compact Riemann surfaces, the collar theorem and Bers’ partition theorem are major tools for working with simple closed geodesics. The main goal of this article is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone angles to be strictly less than π to be able to consider partitions. Emily B. Dryden—partially supported by the US National Science Foundation grant DMS-0306752. Hugo Parlier—supported by the Swiss National Science Foundation grants 21-57251.99 and 20-68181.02.     

Vertex Distinguishing Equitable Total Chromatic Number of Join Graph

A vertex distinguishing equitable total coloring of graph G is a proper total coloring of graph G such that any two distinct vertices＇ coloring sets are not identical and the difference of the elements colored by any two colors is not more than 1. In this paper we shall give vertex distinguishing equitable total chromatic number of join graphs Pn VPn, Cn VCn and prove that they satisfy conjecture 3, namely, the chromatic numbers of vertex distinguishing total and vertex distinguishing equitable total are the same for join graphs Pn V Pn and Cn ∨ Cn.     

Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.     

Block separation in solvable groups

If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π. Received: 22 December 2004     

PACKING CIRCLES OF TWO DIFFERENT SIZES ON THE SPHERE. II

The authors prove that the density of any packing of disks with radii r ₅ = π/5 and R ₅ = π/2 − π/5 cannot exceed the density of the incircles of the Archimedean tessellation of symbol (4,4,5). This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Modular Version of the Gluck-Wolf Theorem

Let N be a normal subgroup of a finite group G. Let ϕ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/ϕ(1) is a π′-number of any χ∈IBr _p (G/ϕ). If p∤|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If p∉π then G/N has a metabelian Hall-π subgroup. Received February 22, 2000, Accepted May 9, 2001