Fractional f-Edge Cover Chromatic Index of Graphs

Let G be a graph, and let f be an integer function on V with ${1\leq f(v)\leq d(v)}$ to each vertex ${\upsilon \in V}$ . An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex ${\upsilon \in V(G)}$ at least f(υ) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by ${\chi^{'}_{fc}(G)}$ . It is well known that any simple graph G has the f-edge cover chromatic index equal to δ _f (G) or δ _f (G) ? 1, where ${\delta_{f}(G)=\,min\{\lfloor \frac{d(v)}{f(v)} \rfloor: v\in V(G)\}}$ . The fractional f-edge cover chromatic index of a graph G, denoted by ${\chi^{'}_{fcf}(G)}$ , is the fractional f-matching number of the edge f-edge cover hypergraph ${\mathcal{H}}$ of G whose vertices are the edges of G and whose hyperedges are the f-edge covers of G. In this paper, we give an exact formula of ${\chi^{'}_{fcf}(G)}$ for any graph G, that is, ${\chi^{'}_{fcf}(G)=\,min \{\min\limits_{v\in V(G)}d_{f}(v), \lambda_{f}(G)\}}$ , where ${\lambda_{f}(G)=\min\limits_{S} \frac{|C[S]|}{\lceil (\sum\limits_{v\in S}{f(v)})/2 \rceil}}$ and the minimum is taken over all nonempty subsets S of V(G) and C[S] is the set of edges that have at least one end in S.     

Some properties onf-edge covered critical graphs

LetG(V, E) be a simple graph, and letf be an integer function onV with 1 ≤f(v) ≤d(v) to each vertexv ∈V. An f-edge cover-coloring of a graphG is a coloring of edge setE such that each color appears at each vertexv ∈V at leastf(v) times. Thef-edge cover chromatic index ofG, denoted by χ′ _fc (G), is the maximum number of colors such that anf-edge cover-coloring ofG exists. Any simple graphG has anf-edge cover chromatic index equal to δ_f or δ _f - 1, where $\delta _f = \mathop {\min }\limits_{\upsilon \in V} \{ \left\lfloor {\frac{{d(v)}}{{f(v)}}} \right\rfloor \} $ . LetG be a connected and not complete graph with χ′ _fc (G)=δ _f-1, if for eachu, v ∈V and e =uv ?E, we have ÷ _fc (G + e) > ÷ _fc (G), thenG is called anf-edge covered critical graph. In this paper, some properties onf-edge covered critical graph are discussed. It is proved that ifG is anf-edge covered critical graph, then for eachu, v ∈V and e =uv ?E there existsw ∈ {u, v } withd(w) ≤ δ _f (f(w) + 1) - 2 such thatw is adjacent to at leastd(w) - δ _f + 1 vertices which are all δ _f -vertex inG.     

Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients

The Hardy type inequality $\left( * \right) \left( {\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f\left( k \right)} \right|^p }}{{k^{2 - p} }}} } \right)^{I/p} \leqslant C_p \left\| f \right\|_{H_{ * * }^P } \left( {1/2< p \leqslant 2} \right)$ is proved for functionsf belonging to the Hardy spaceH _** ^p (G_m) defined by means of a maximal function. We extend (*) for 2<p<∞ when the Vilenkin-Fourier coefficients off are λ-blockwise monotone. It will be shown that under certain conditions on the Vilenkin system (in particular, for some unbounded type, too) a converse version of (*) holds also for allp>0 provided that the Vilenkin-Fourier coefficients off are monotone.     

Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {?1, 1} be a two-valued function. If k ≥ 1 is an integer and ${\sum_{x\in N[v]} f(x) \ge k}$ for each ${v \in V(G)}$ , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f ₁,f ₂, . . . ,f _d } of distinct signed k-dominating functions on G with the property that ${\sum_{i=1}^d f_i(x) \le k}$ for each ${x \in V(G)}$ , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.     

On Group Choosability of Graphs,II

Given a group A and a directed graph G, let F(G, A) denote the set of all maps ${f : E(G) \rightarrow A}$ . Fix an orientation of G and a list assignment ${L : V(G) \mapsto 2^A}$ . For an ${f \in F(G, A)}$ , G is (A, L, f)-colorable if there exists a map ${c:V(G) \mapsto \cup_{v \in V(G)}L(v)}$ such that ${c(v) \in L(v)}$ , ${\forall v \in V(G)}$ and ${c(x)-c(y)\neq f(xy)}$ for every edge e = xy directed from x to y. If for any ${f\in F(G,A)}$ , G has an (A, L, f)-coloring, then G is (A, L)-colorable. If G is (A, L)-colorable for any group A of order at least k and for any k-list assignment ${L:V(G) \rightarrow 2^A}$ , then G is k-group choosable. The group choice number, denoted by ${\chi_{gl}(G)}$ , is the minimum k such that G is k-group choosable. In this paper, we prove that every planar graph is 5-group choosable, and every planar graph with girth at least 5 is 3-group choosable. We also consider extensions of these results to graphs that do not have a K ₅ or a K _3,3 as a minor, and discuss group choosability versions of Hadwiger’s and Woodall’s conjectures.     

Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups

Let (χ, G, U) be a continuous representation of a Lie groupG by bounded operatorsg →U(g) on the Banach space χ and let (χ, $\mathfrak{g}$ ,dU) denote the representation of the Lie algebra $\mathfrak{g}$ obtained by differentiation. Ifa ₁, ...,a _d′ is a Lie algebra basis of $\mathfrak{g}$ ,A _i=dU(a _i) and $A^\alpha = A_{i_1 } ...A_{i_k } $ whenever α=(i ₁, ...,i _k) we consider the operators $$H = \mathop \sum \limits_{\alpha ;|\alpha | \leqslant 2n} c_\alpha A^\alpha $$ where thec _α are complex coefficients satisfying a subcoercivity condition. This condition is such that the class of operators considered encompasses all the standard second-order subelliptic operators with real coefficients, all operators of the form $\sum _{i = 1}^{d'} \lambda _i ( - A_i^2 )^n $ with Re λ _i >0 together with operators of the form $$H = ( - 1)^n \mathop \sum \limits_{\alpha ;|\alpha | = n} \mathop \sum \limits_{\beta ;|\beta | = n} c_{\alpha ,\beta } A^{\alpha _* } A^\beta $$ where α_*=(i _k, ...,i ₁) if α=(i ₁, ...,i _k) and the real part of the matrix (c _{α β}) is strictly positive. In case the Lie algebra $\mathfrak{g}$ is free of stepr, wherer is the rank of the algebraic basisa ₁, ...,a _d′,G is connected andU is the left regular representation inG we prove that the closure $\overline H $ ofH generates a holomorphic semigroupS. Moreover, the semigroupS has a smooth kernel and we derive bounds on the kernel and all its derivatives. This will be a key ingredient for the paper [13] in which the above results will be extended to general groups and representations.     

The Signed k-Submatchings in Graphs

A signed k-submatching of a graph G is a function f : E(G) → {?1,1} satisfying f (E _G (v)) ≤ 1 for at least k vertices ${v \in V(G)}$ . The maximum of the values of f (E(G)), taken over all signed k-submatchings f, is called the signed k-submatching number and is denoted by ${\beta_S^{k}(G)}$ . In this paper, sharp bounds on ${\beta_S^{k}(G)}$ for general graphs are presented. Exact values of ${\beta_S^{k}(G)}$ for several classes of graphs are found.     

The {k}-domatic number of a graph

For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex ${v\in V(G)}$ , the condition ${\sum_{u\in N[v]}f(u)\ge k}$ is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f ₁, f ₂, . . . , f _d } of {k}-dominating functions on G with the property that ${\sum_{i=1}^df_i(v)\le k}$ for each ${v\in V(G)}$ , is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d _{k}(G). Note that d _{1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d _{k}(G). Many of the known bounds of d(G) are immediate consequences of our results.     

A relationship between bounds on the sum of squares of degrees of a graph

LetG = (V, E) be a simple graph withn vertices and e edges. Letdi be the degree of the ith vertex v_i ∈ V andm _i the average of the degrees of the vertices adjacent to vertexv _i ∈ V. It is known by Caen [1] and Das [2] that $\frac{{4e^2 }}{n} \leqslant d_1^2 + ... + d_n^2 \leqslant e max \{ d_j + m_j |v_j \in V\} \leqslant e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ . In general, the equalities do not hold in above inequality. It is shown that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 $ . In fact, it is shown a little bit more strong result that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} $ . For a graphG withn < 2 vertices, it is shown that G is a complete graphK _n if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} = e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ .     

Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings

Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ₁ and Δ₂ from ${R^n \to R}$ such that Δ₁(δ ₂(x), x, . . . ,x) =  0 for all ${x \in R,}$ where δ ₂ is the trace of Δ₂. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ ₁, δ ₂, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ ₁ δ ₂(x) =  f(x) holds for all ${x \in R}$ , then R is commutative.     

Bounds on the 2-Rainbow Domination Number of Graphs

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any ${v\in V(G), f(v)=\emptyset}$ implies ${\bigcup_{u\in N(v)}f(u)=\{1,2\}.}$ The 2-rainbow domination number γ _r2(G) of a graph G is the minimum ${w(f)=\Sigma_{v\in V}|f(v)|}$ over all such functions f. Let G be a connected graph of order |V(G)| = n ≥ 3. We prove that γ _r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ _r2(G) in terms of diameter are also given.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Exactn-widths of hardy-sobolev classes

Let $\tilde h^r _{\infty ,\beta } $ and $\tilde H^r _{\infty ,\beta } $ denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref ^(r)(z)| ≤ 1 and |f ^(r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of $\tilde h^r _{\infty ,\beta } $ inL _q[0, 2π], 1 ≤q ≤ ∞, and $\tilde H^r _{\infty ,\beta } $ inL _∞[0, 2π].     

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.     

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 ${\mathbb{N} - \{2,4\}.}$      

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

The Linear t-Colorings of Sierpiński-Like Graphs

A proper t-coloring of a graph G is a mapping ${\varphi: V(G) \rightarrow [1, t]}$ such that ${\varphi(u) \neq \varphi(v)}$ if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by ${\chi(G)}$ , is the minimum number of colors required in any proper coloring of G. A linear t-coloring of a graph is a proper t-coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by ${lc(G)}$ , is the minimum t such that G has a linear t-coloring. In this paper, the linear t-colorings of Sierpiński-like graphs S(n, k), ${S^+(n, k)}$ and ${S^{++}(n, k)}$ are studied. It is obtained that ${lc(S(n, k))= \chi (S(n, k)) = k}$ for any positive integers n and k, ${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$ and ${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$ for any positive integers ${n \geq 2}$ and ${k \geq 3}$ . Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely.     

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.