The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Compact AC(σ) Operators

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts.      

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Fourier algebras on homogeneous spaces

Spectral synthesis and operator synthesis on a homogeneous space G/K, where K is a compact subgroup of a locally compact group G, are studied. Injection theorem for sets of spectral synthesis for A(G/K) is proved, extending the classical result of Reiter and more recent results of Kaniuth–Lau, Parthasarathy–Prakash and others. A simple direct image theorem for spectral synthesis is proved and an extension of the subgroup theorem and an alternate proof of the injection theorem are obtained as consequences. The relation between synthesis in the Fourier algebra A(G/K) and an appropriate Varopoulos algebra is obtained, subsuming earlier results of Varopoulos, Spronk–Turowska and Parthasarathy–Prakash. Study of relations between spectral synthesis and operator synthesis pioneered by Arveson and carried forward recently by Shulman–Turowska, Parthasarathy–Prakash and Ludwig–Turowska is undertaken on homogeneous spaces. Operator space methods are needed for this study, and more specifically, a characterisation of completely bounded multipliers on A(G/K) as the invariant part of a suitable weak^? Haagerup tensor product (or the space of Schur multipliers) is given and is used for this study.     

Homological properties of Fourier algebras on homogeneous spaces

Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A(G/K) are characterised. Biprojectivity and biflatness for the Fourier algebra A(G/K) are studied. A(G/K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup.     

A Class of Operators Similar to the Shift on H 2(G)

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that and every function in H^∞(G) is the pointwise limit of a bounded sequence of functions in . The purpose of this paper is to characterize all subnormal operators similar to M_z, the operator of multiplication by the independent variable z on the Hardy space H²(G). We also characterize all bounded linear operators that are unitarily equivalent to M_z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.     

The co-degrees of irreducible characters

LetG be a finite group. The co-degree of an irreducible character χ ofG is defined to be the number |G|/χ(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters ofG is denoted by Σ(G). First we show that Σ(G)=π(G) (the set of all prime divisors of |G|) unlessG is nilpotent-by-abelian. Then we make Σ(G) a graph by adjoining two elements of Σ(G) if and only if their product divides a co-degree of some nonlinear character ofG. We show that the graph Σ(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that ifG has a co-degree which is ap-power for some primep, then the corresponding character is monomial andO _p (G)≠1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well. The paper was written during this author’s visit at the Technion and the University of Tel Aviv. He would like to thank the departments of mathematics at the Technion and the University of Tel Aviv for their hospitality and support.     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

Automorphisms of blocks of group algebras and character values

Let G be a finite group, χ an irreducible complex character of G and A(χ) the block ideal of the group algebra ℚG relatedℴ χ. The aim of this paper is to study the group Aut (A(χ)) of all ring (or ℚ-algebra) automorphisms of A(χ). Especially we are interested in the existence of subgroups of Aut (A(χ)), which are isomorphic to a given subgroup Γ of the Galois group of the field of character values ℚ(χ) over the rationals. In this context we prove some results related to character values.     

On zeros of characters of finite groups

For a finite group G and a non-linear irreducible complex character χ of G write υ(χ) = {g ∈ G | χ(g) = 0}. In this paper, we study the finite non-solvable groups G such that υ(χ) consists of at most two conjugacy classes for all but one of the non-linear irreducible characters χ of G. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable φ-groups. As a corollary, we answer Research Problem 2 in [Y.Berkovich and L.Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math. 24 (1998), 619–630.] posed by Y.Berkovich and L.Kazarin.     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

The Entire Coloring of Series-Parallel Graphs

The entire chromatic number X_(vef)(G) of a plane graph G is the minimal number of colors needed for coloring vertices, edges and faces of G such that no two adjacent or incident elements are of the same color. Let G be a series-parallel plane graph, that is, a plane graph which contains no subgraphs homeomorphic to K_(4-) It is proved in this paper that X_(vef)(G)≤max{8, △(G) 2} and X_(vef)(G)=△ 1 if G is 2-connected and △(G)≥6.     

Total colorings of graphs of order 2n having maximum degree 2n−2

Let χ _t (G) and †(G) denote respectively the total chromatic number and maximum degree of graphG. Yap, Wang and Zhang proved in 1989 that ifG is a graph of orderp having †(G)≥p−4, then χ _t (G≤Δ(G)+2. Hilton has characterized the class of graphG of order 2n having †(G)=2n−1 such that χ _t (G=Δ(G)+2. In this paper, we characterize the class of graphsG of order 2n having †(G)=2n−2 such that χ _t (G=Δ(G)+2 Research supported by National Science Council of the Republic of China (NSC 79-0208-M009-15)     

Fractional and integral colourings

LetG=(V, E) be an undirected graph andc any vector in ℤ ^V(G) ₊. Denote byχ(G _c) (resp.η(G _c)) the chromatic number (resp. fractional chromatic number) ofG with respect toc. We study graphs for whichχ(G _c)−[η(G _c)]⩽1. We show that for the class of graphs satisfyingχ(G _c)=[η(G _c)] (a class generalizing perfect graphs), an analogue of the Duplication Lemma does not hold. We also describe a 2-vertex cut decomposition procedure related to the integer decomposition property. We use this procedure to show thatχ(G _c)=[η(G _c)] for series-parallel graphs andχ(G _c)⩽[η(G _c)]+1 for graphs that do not have the 4-wheel as a minor. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC) under grant A9126.     

Analytic actions on compact surfaces and fixed points

 Consider an effective real analytic action of a connected Lie group G on a compact connected surface of Euler characteristic χ≠0. We show that if the action has no fixed point then χ≥1 and the Lie algebra 𝒢 of G is isomorphic either to a subalgebra of the affine algebra of ℝ², which is the extension of the ideal of constant vector fields by an irreducible linear subalgebra, or to sl(2,ℝ), o(3), sl(2,ℂ) and sl(3,ℝ). Received: 7 August 2001 Published online: 24 January 2003