Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth

We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a ${\overleftrightarrow{K_4}}$ -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here ${\overleftrightarrow{K_n}}$ means a digraph with n vertices and arcs in both directions between every distinct pair), and a ${\overleftrightarrow{K_3}}$ -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.     

More on monochromatic-rainbow Ramsey type theorems

An edge colored graph is called a rainbow if no two of its edges have the same color. Let ? and $\mathcal{G}$ be two families of graphs. Denote by $RM({\mathcal{H}},\mathcal{G})$ the smallest integer R, if it exists, having the property that every coloring of the edges of K _R by an arbitrary number of colors implies that either there is a monochromatic subgraph of K _R that is isomorphic to a graph in ? or there is a rainbow subgraph of K _R that is isomorphic to a graph in $\mathcal{G}$ . ${\mathcal{T}}_{n}$ is the set of all trees on n vertices. ${\mathcal{T}}_{n}(k)$ denotes all trees on n vertices with diam(T _n (k))≤k. In this paper, we investigate $RM({\mathcal{T}}_{n},4K_{2})$ , $RM({\mathcal{T}}_{n},K_{1,4})$ and $RM({\mathcal{T}}_{n}(4),K_{3})$ .     

On the Broadcast Independence Number of Grid Graph

A broadcast on a nontrivial connected graph G is a function ${f:V \longrightarrow \{0, \ldots,\operatorname{diam}(G)\}}$ such that for every vertex ${v \in V(G)}$ , ${f(v) \leq e(v)}$ , where ${\operatorname{diam}(G)}$ denotes the diameter of G and e(v) denotes the eccentricity of vertex v. The broadcast independence number is the maximum value of ${\sum_{v \in V} f(v)}$ over all broadcasts f that satisfy ${d(u,v) > \max \{f(u), f(v)\}}$ for every pair of distinct vertices u, v with positive values. We determine this invariant for grid graphs ${G_{m,n} = P_m \square P_n}$ , where ${2 \leq m \leq n}$ and □ denotes the Cartesian product. We hereby answer one of the open problems raised by Dunbar et al. in (Discrete Appl Math 154:59–75, 2006).     

Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

Let G=(V,E) be a finite, simple graph. We consider for each oriented graph $G_{\mathcal{O}}$ associated to an orientation ${\mathcal{O}}$ of the edges of G, the toric ideal $P_{G_{\mathcal{O}}}$ . In this paper we study those graphs with the property that $P_{G_{\mathcal{O}}}$ is a binomial complete intersection, for all ${\mathcal{O}}$ . These graphs are called $\text{CI}{\mathcal{O}}$ graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce chorded-theta subgraphs and some of their properties. Also we establish that the $\text{CI}{\mathcal{O}}$ graphs are determined by the property that each chorded-theta has a transversal triangle. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of forbidden graphs are: prisms, pyramids, thetas and a particular family of wheels that we call θ-partial wheels.     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

The Majorization Theorem for Signless Laplacian Spectral Radii of Connected Graphs

Let ${\pi=(d_{1},d_{2},\ldots,d_{n})}$ and ${\pi'=(d'_{1},d'_{2},\ldots,d'_{n})}$ be two non-increasing degree sequences. We say ${\pi}$ is majorizated by ${\pi'}$ , denoted by ${\pi \vartriangleleft \pi'}$ , if and only if ${\pi\neq \pi'}$ , ${\sum_{i=1}^{n}d_{i}=\sum_{i=1}^{n}d'_{i}}$ , and ${\sum_{i=1}^{j}d_{i}\leq\sum_{i=1}^{j}d'_{i}}$ for all ${j=1,2,\ldots,n}$ . If there exists one connected graph G with ${\pi}$ as its degree sequence and ${c=(\sum_{i=1}^{n}d_{i})/2-n+1}$ , then G is called a c-cyclic graph and ${\pi}$ is called a c-cyclic degree sequence. Suppose ${\pi}$ is a non-increasing c-cyclic degree sequence and ${\pi'}$ is a non-increasing graphic degree sequence, if ${\pi \vartriangleleft \pi'}$ and there exists some t ${(2\leq t\leq n)}$ such that ${d'_{t}\geq c+1}$ and ${d_{i}=d'_{i}}$ for all ${t+1\leq i\leq n}$ , then the majorization ${\pi \vartriangleleft \pi'}$ is called a normal majorization. Let μ(G) be the signless Laplacian spectral radius, i.e., the largest eigenvalue of the signless Laplacian matrix of G. We use C _π to denote the class of connected graphs with degree sequence π. If ${G \in C_{\pi}}$ and ${\mu(G)\geq \mu(G')}$ for any other ${G'\in C_{\pi}}$ , then we say G has greatest signless Laplacian radius in C _π . In this paper, we prove that: Let π and π′ be two different non-increasing c-cyclic (c ≥ 0) degree sequences, G and G′ be the connected c-cyclic graphs with greatest signless Laplacian spectral radii in C _π and C _π', respectively. If ${\pi \vartriangleleft \pi'}$ and it is a normal majorization, then ${\mu(G) < \mu(G')}$ . This result extends the main result of Zhang (Discrete Math 308:3143–3150, 2008).     

Covering functions by countably many functions from some families

Let $ \mathcal{A} $ be a nonempty family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence (f _n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr(f) denotes the graph of f). If $ \mathcal{A} $ is the family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.     

On the signed star domination number of regular multigraphs

Let $G$ be a graph with the vertex set $V(G)$ and the edge set $E(G)$ . A function $f: E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star dominating function of $G$ if $\sum _{e \in E_G(v)}f (e)\ge 1 $ , for every $v \in V(G)$ , where $E_G(v) = \{uv\in E(G)\,|\,u \in V (G)\}$ . The minimum values of $\sum _{e \in E_G(v)}f (e)$ , taken over all signed star dominating functions $f$ on $G$ , is called the signed star domination number of $G$ and denoted by $\gamma _{SS}(G)$ . In this paper we determine the signed star domination number of regular multigraphs.     

Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), ${\Lambda}$ a k-dimensional topological manifold with topological orientation ${\eta}$ , and ${f : D \rightarrow X}$ a locally compact map, where D is an open subset of ${X \times \Lambda}$ . We define Fix(f) as the set of points ${{(x, \lambda) \in D}}$ such that ${x = f(x, \lambda)}$ . For an open pair (U, V) in ${X \times \Lambda}$ such that ${{\rm Fix}(f) \cap U \backslash V}$ is compact we construct a homomorphism ${\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}$ in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a ${C^{\infty}}$ -manifold ${\Lambda}$ , these properties uniquely determine ${\Sigma}$ . By passing to the direct limit of ${\Sigma_{(f,U,V)}}$ with respect to the pairs (U, V) such that ${K = {\rm Fix}(f) \cap U \backslash V}$ , we define a homomorphism ${\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}$ in the ?ech cohomologies. Properties of ${\Sigma}$ and ${\sigma}$ are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.     

On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

A Note on Classical and p-adic Fréchet Functional Equations with Restrictions

Given X,Y two ${\mathbb{Q}}$ -vector spaces, and f : X → Y, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$ , if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$ , then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$ .     

The Chvátal–Erdös Condition for Group Connectivity in Graphs

Let G be a graph and A an abelian group with the identity element 0 and ${|A| \geq 4}$ . Let D be an orientation of G. The boundary of a function ${f: E(G) \rightarrow A}$ is the function ${\partial f: V(G) \rightarrow A}$ given by ${\partial f(v) = \sum_{e \in E^+(v)}f(e) - \sum_{e \in E^-(v)}f(e)}$ , where ${v \in V(G), E^+(v)}$ is the set of edges with tail at v and ${E^-(v)}$ is the set of edges with head at v. A graph G is A-connected if for every b: V(G) → A with ${\sum_{v \in V(G)} b(v) = 0}$ , there is a function ${f: E(G) \mapsto A-\{0\}}$ such that ${\partial f = b}$ . A graph G is A-reduced to G′ if G′ can be obtained from G by contracting A-connected subgraphs until no such subgraph left. Denote by ${\kappa^{\prime}(G)}$ and α(G) the edge connectivity and the independent number of G, respectively. In this paper, we prove that for a 2-edge-connected simple graph G, if ${\kappa^{\prime}(G) \geq \alpha(G)-1}$ , then G is A-connected or G can be A-reduced to one of the five specified graphs or G is one of the 13 specified graphs.     

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .