Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008     

Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.     

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G)and an edge set E(G). A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1}.The weight of f is W(f)=∑_(x∈V)(G)∪E(G))f(X).For an element x∈V(G)∪E(G),we define f[x]=∑_(y∈NT[x])f(y).A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1} such that f[x]≥1 for all x∈V(G)∪E(G).The total signed domination numberγ_s~*(G)of G is the minimum weight of a total signed domination function on G. In this paper,we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofγ_s~*(G)when G is C_n and P_n.     

Piecewise Endomorphisms of Pid-Modules

Let G be a finitely generated module over a PID, D. We investigate the structure of the centralizer near-ring M_D(G) = {f: G → G ¦ f(ar) = (fa)r,a ∈ G, r ∈ D}. If C = {G_α} is a cover of G by maximal cyclic submodules then we show that every f ∈ M_D(G) is piecewise an endomorphism of G.     

Maximal subrings and <Emphasis Type="Italic">E</Emphasis>-groups

For a finite group G, let E(G) denote the near-ring of functions generated by the semigroup, End(G), of endomorphisms of G. We characterize when E(G) is maximal as a subnear-ring of M ₀(G). A group G is an E-group if E(G) is a ring. We give a new characterization of finite E-groups and investigate the problem of determining, for finite E-groups, when E(G) is maximal as a ring in M₀(G). Received: 26 June 2006     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

<Emphasis Type="Italic">f</Emphasis>-Colorings of some graphs of <Emphasis Type="Italic">f</Emphasis>-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G） ＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.     

The Roman <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f ₁, f ₂, …, f _d } of distinct Roman k-dominating functions on G with the property that Σ _i=1 ^d f _i (v) ≤ 2 for each v ∈ V (G), is called a Roman k-dominating family (of functions) on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by d _kR (G). Note that the Roman 1-domatic number d _1R (G) is the usual Roman domatic number d _R (G). In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for d _kR (G). In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.     

Topological rings,homogeneous functions,and primeness

For any group G such that G is a right R-module for some ring R, the elements of R act on G as endomorphisms and we obtain the near-ring of R-homogeneous maps on G: M_R(G) = {f: G → G|f(ga) = f(g)a for all a ∈ R, g ∈ G}. In the special case that R is a topological ring and G is a topological R-module, we study N_R(G): = {f ∈ M_R(G)|f is continuous}. In particular, we investigate primeness of the near-ring N_R(G) of continuous homogeneous maps on G.     

T-Colorings and T-Edge Spans of Graphs

 Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C ^d _n of the n-cycle C _n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C _n ^d for n≡0 or (mod d+1), and lower and upper bounds for other cases. Received: May 13, 1996 Revised: December 8, 1997     

Minus total domination in graphs

A three-valued function f: V → {−1, 0, 1} defined on the vertices of a graph G= (V, E) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every υ ∈ V, f(N(υ)) ⩾ 1, where N(υ) consists of every vertex adjacent to υ. The weight of an MTDF is f(V) = Σf(υ), over all vertices υ ∈ V. The minus total domination number of a graph G, denoted γ _t ⁻(G), equals the minimum weight of an MTDF of G. In this paper, we discuss some properties of minus total domination on a graph G and obtain a few lower bounds for γ _t ⁻(G).     

Factors and Connected Induced Subgraphs

 Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg _G (x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑ _{x ∈ V ( G )} f(x) is odd. Received: June 29, 1998?Final version received: July 30, 1999     

Hypercyclic Pairs of Coanalytic Toeplitz Operators

A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A ⁿ B ^k x : n, k ≥ 0} is dense in . If f, g ∈H ^∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M ^* _f , M ^* _g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H ^∞(G) such that the pair (M ^* _f , M ^* _g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.     

Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

On the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V(G)) = ?_{u ? V(G)} f (u){f (V(G)) = \sum_{u\in V(G)} f (u)}. The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b _R (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E^￠ í E(G){E^{\prime} \subseteq E(G)} for which γ _R (G − E′) > γ _R (G). In this paper we present different bounds on the Roman bondage number of planar graphs.     

Unique maximal rings of functions

For several classes of groups G, we characterize when the near-ring M₀(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M₀(G) are commutative.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

Bounds for regularity and coregularity of graded modules

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R ₀, m₀). If R ₀ is of dimension one, then we show that reg ⁱ⁺¹(M) and coreg ⁱ⁺¹(M) are bounded for all i ∈ ℕ₀. We improve these bounds, if in addition, R ₀ is either regular or analytically irreducible of unequal characteristic.