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This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let \(\left\{ {X_a } \right\}_{a \in Z_ + }\) be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by L n the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X 1, ..., X n . In this paper, we obtain a limit law and a week version of an Erdos-Renyi type law for L n . A large deviation result of L n is also discussed.

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be \(O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)\). This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

Plesnik in 1972 proved that an (m − 1)-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m − 1 edges. Alder et al. in 1999 showed that if G is a regular (2n + 1)-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and (or) excluding a set of edges, where 1 ≤ k ≤ n/2. In particular, we generalize Plesnik’s result and the results obtained by Liu et al. in 1998, and improve Katerinis’ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.

An excellent introduction to the topic of poset matroids is due to M. Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rank axioms for poset matroids. In this paper, we study the special integral function f and obtain a new class of poset matroids from the old ones, and then we generalize this result according to the properties of f. Almost all of these results can be regarded as the application of global rank axioms for poset matroids. The main results in our paper have, indeed, investigated the restriction of the basis of the poset matroid, and we give them the corresponding geometric interpretation.

Let G = (V,E) be a simple graph. A function f: E → {+1,−1} is called a signed cycle domination function (SCDF) of G if Ʃ e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{Ʃ e∈E f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ′sc(G) = n.

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < $$\tfrac{{37}} {{12}}$$ and Δ(G) ≥ 7.

Let S = (a 1, …, a m ; b 1, …, b n ), where a 1, …, a m and b 1, …, b n are two nonincreasing sequences of nonnegative integers. The pair S = (a 1, …, a m ; b 1, …, b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y, E) such that a 1, …, a m and b 1, …, b n are the degrees of the vertices in X and Y, respectively. Let Z 3 be the cyclic group of order 3. Define σ(Z 3, m, n) to be the minimum integer k such that every bigraphic pair S = (a 1, …, a m ; b 1, …, b n ) with a m , b n ≥ 2 and σ(S) = a 1 + ⋯ + a m ≥ k has a Z 3-connected realization. For n = m, Yin [Discrete Math., 339, 2018—2026 (2016)] recently determined the values of σ(Z 3, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z 3, m, n) for m ≥ n ≥ 4.

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different fromthe sumof colors taken on the edges incident to v. Let χ′Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) < 5/2, then χ′Σ(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.

In this paper, we investigate the intersection numbers of nearly Kirkman triple systems. JN[v] is the set of all integers k such that there is a pair of NKTS(v)s with a common uncovered collection of 2-subset intersecting in k triples. It has been established that \({J_N}\left[ v \right] = \left\{ {0,1,...,\frac{{v\left( {v - 2} \right)}}{6} - 6,\frac{{v\left( {v - 2} \right)}}{6} - 4,\frac{{v\left( {v - 2} \right)}}{6}} \right\}\) for any integers v = 0 (mod 6) and v ≥ 66. For v ≤ 60, there are 8 cases left undecided.

Let X denote a compact metric space with distance d and F : X × ℝ → X or Ft : X → X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U = (1, 1), down steps D = (1,−1), horizontal steps H = (1, 0), and left steps L = (−1,−1), and such that up steps never overlap with left steps. Let S n be the set of all skew Motzkin paths of length n and let s n = |S n |. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {s n } n ≥0. Then we present several involutions on S n and consider the number of their fixed points. Finally we consider the enumeration of some statistics on S n .

In this paper, the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied. There are three approaches to solve this problem. The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths; the second approach is to find a current assignment of the current graph by the theory of current graph; the third approach is to find exponentially many embedding (or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph. According to this three approaches, we can construct exponentially many minimum genus embeddings of complete graph K 12s+8 in orientable surfaces, which show that there are at least $$\frac{10}{3} \times (\frac{200}{9})^s$$ distinct minimum genus embeddings for K 12s+8 in orientable surfaces. We have also proved that K 12s+8 has at least $$\frac{10}{3} \times (\frac{200}{9})^s$$ distinct minimum genus embeddings in non-orientable surfaces.

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A gc-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a gc-coloring with k colors is called the gc-chromatic index of G and denoted by \(\chi\prime_{g_{c}}\) (G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\) (G) = δg(G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).

Let G be a finite group, Irr1(G) be the set of nonlinear irreducible characters of G and cd1(G) the set of degrees of the characters in Irr1(G). A group G is said to be a D2-group if |cd1(G)| = |Irr1(G)| − 2. In this paper, we give a complete classification of solvable D2-groups.

We first determine in this paper the structure of the generalized Fitting subgroup F* (G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/O p (L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O 2(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color. In this paper, it is proved that if G is a planar graph with Δ(G) ≥ 7 and without chordal 7-cycles, then G has a (Δ(G)+1)-total-coloring.

Let Π = B1/B2/ ··· /Bk be any set partition of [n] = {1, 2,..., n} satisfying that entries are increasing in each block and blocks are arranged in increasing order of their first entries. Then Callan defined the flattened Π to be the permutation of [n] obtained by erasing the divers between its blocks, and Callan also enumerated the number of set partitions of [n] whose flattening avoids a single 3-letter pattern. Mansour posed the question of counting set partitions of [n] whose flattening avoids a pattern of length 4. In this paper, we present the number of set partitions of [n] whose flattening avoids one of the patterns: 1234, 1243, 1324, 1342, 1423, 1432, 3142 and 4132.

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ′ a (G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ′ a (G) ≤ Δ(G)+22, if G is a triangle-free 1-planar graph.

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V (G), then G contains a cycle of weight at least 2d.

Let G be a finite centerless group, let π(G) be the set of prime divisors of the order of G, and let np(G) be the number of Sylow p-subgroups of G, that is, np(G) = |Sylp(G)|. Set NS(G):= {np(G)| p ∈ π(G)}. In this paper, we are investigating whether L2(r) is determined up to isomorphism by NS(L2(r)) when r is prime.

By Petersen’s Theorem, a bridgeless cubic graph has a 2-factor. Fleischner (Discrete Math., 101, 33–37 (1992)) has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let me > 0 be even and mo > 0 be odd. In this paper, we prove that every me-edge-connected graph with minimum degree at least me + 1 contains an even factor with minimum degree at least me and every (mo + 1)-edge-connected graph contains an odd factor with minimum degree at least mo, which further extends Fleischner’s result. Moreover, we show that our results are best possible.

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph (Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph (also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits (of equal size). In this paper, we introduce the concept of SCI-graph (semi-Cayley isomorphism) and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 > q2 > 1, the algorithm has \(O((q_1 + 1)n^{\frac{{q_1 + 1}} {{2(q_1 - q_2 )}}} \log \tfrac{n} {e}) \) and \(O((q_1 + 1)^{\frac{{3q_1 - 2q_2 + 1}} {{2(q_1 - q_2 )}}} \sqrt n \log \tfrac{n} {e}) \) complexity results for large- and small-update methods, respectively.

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.

A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, ..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) ≠ f(v). By χ″nsd(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured that χ″nsd(G) ≤ Δ(G)+3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.

In this paper, the cycle structures for directed graphs on surfaces are studied. If G is a strongly connected graph, C is a Π-contractible directed cycle of G, then both of Int(C,Π) and Ext(C,Π) are strongly connected graph; the dimension of cycles space of G is identified. If G is a strongly connected graph, then the structure of MCB in G is unique. Let G be a strongly connected graph, if G has been embedded in orientable surface Sg with fw(G) ≥ 2 (fw(G) is the face-width of G), then any cycle base of G must contain at least 2g noncontractible directed cycles; if G has been embedded in non-orientable surface Ng, then any cycle base of G must contain at least g noncontractible directed cycles.

Let θ ∈ ℝ d be a unit vector and let X,X 1,X 2, … be a sequence of i.i.d. ℝ d -valued random vectors attracted to operator semi-stable laws. For each integer n ≥ 1, let X 1,n ≤ … ≤ X n,n denote the order statistics of X 1,X 2, …, X n according to priority of index, namely |〈X 1,n , θ〉| ≥ … ≥ |〈X n,n , θ〉|, where 〈·, ·〉 is an inner product on ℝ d . For all integers r ≥ 0, define by (r) S n = Σ =1 X i,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums (r) S n . Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of (r) S n is obtained.

A digraph D is oriented if it does not contain 2-cycles. If an oriented digraph D has a directed eulerian path, it is an oriented eulerian digraph. In this paper, when an oriented eulerian digraph D has minimum out-degree 2 and a diameter d, we find the minimum order of D. In addition, when D is 2-regular with diameter 4m (m ≥ 2), we classify the extremal cases.

In this paper, we consider the family of generalized Petersen graphs P(n, 4). We prove that the metric dimension of P(n, 4) is 3 when n ≡ 0 (mod 4), and is 4 when n = 4k + 3 (k is even). For n ≡ 1,2 (mod 4) and n = 4k + 3 (k is odd), we prove that the metric dimension of P(n, 4) is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P(n, 4) has constant metric dimension.

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, …, h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge u v ∈ E(G), if w(u) ≠ w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndiΣ (G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndiΣ (G) ≤ k + 2 where k = max{Δ(G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak.

Let G (n, k, t) be a set of graphs with n vertices, k cut edges and t cut vertices. In this paper, we classify these graphs in G (n, k, t) according to cut vertices, and characterize the extremal graphs with the largest spectral radius in G (n, k, t).

Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1 − E(G) be the graph obtained from Kr+1 by deleting all edges of G. A non-increasing sequence π = (d1, d2, ..., dn) of nonnegative integers is said to be potentially Kr+1 − E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1−E(G) as a subgraph. In this paper, a characterization of π that is potentially Kr+1 − E(G)-graphic is given, which is analogous to the Erd−os-Gallai characterization of graphic sequences using a system of inequalities. This is a solution to an open problem due to Lai and Hu. As a corollary, a characterization of π that is potentially Ks,t-graphic can also be obtained, where Ks,t is the complete bipartite graph with partite sets of size s and t. This is a solution to an open problem due to Li and Yin.

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper, we show that if G is a graph of order n ≥ 4, then \(\gamma _r \left( G \right)\gamma _r \left( {\bar G} \right) \leqslant 2n\). We also characterize the graphs achieving the upper bound.

A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79–84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph. This extends a former result of Catlin and Lai [J. Combin. Theory, Ser. B, 66, 123–139 (1996)].

A subset system Z assigns to each partially ordered set P a certain collection Z(P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Ƶ, the concepts of FƵ-way-below relation and FƵ-domain are introduced. The well-known Scott topology is naturally generalized to the Ƶ-level and the resulting topology is called FƵ-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Ƶ-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS of FS-domains to the Ƶ-level. Corresponding to them, it is proved that, for a suitable subset system Ƶ, the categories F Ƶ CPO of Ƶ-complete posets, FSF Ƶ of finitely separated FƵ-domains and BFF Ƶ of bifinite FƵ-domains are all cartesian closed. Some examples of these categories are given.

Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zν relative to subgroup of order k, where ν = km is a composite number. As applications, we obtain some new optimal DSSs.

An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1, 2, …, k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is the sum of its label and the labels of its two end vertices. This concept was introduce by Brandt et al. They defined χt′(G) to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question: Is there a constant K with \(\chi '_t (G) \leqslant \tfrac{{\Delta (G) + 1}} {2} + K \) for all graphs G of maximum degree Δ(G)? In this paper, we give a positive answer for outerplanar graphs by showing that \(\chi '_t (G) \leqslant \left\lceil {\tfrac{{\Delta (G) + 1}} {2}} \right\rceil + 1 \) for each outerplanar graph G with maximum degree Δ(G).

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.: Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17(2), 493–502 (2004)].

In this paper, we consider a discrete version of Aleksandrov’s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice’s y-coordinates’ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov’s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λ c , if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549–562 (2011)], it is proved that a connected vertex-transitive graph is super-λ c if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λ c graphs which have degree 4 and girth 5. In this paper, a characterization of k (k ≥ 4)-regular vertex-transitive nonsuper-λ c graphs of girth 5 is given. Using this, we classify all k (k ≥ 4)-regular nonsuper-λ c Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λ c vertex-transitive non-Cayley graphs.

Let G be a planar graph with maximum degree Δ. In this paper, we prove that if any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4} in G, then the list edge chromatic number x l ′ (G) = Δ and the list total chromatic number x l ″ (G) = Δ+1.

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τC(G), is the minimum cardinality of a clique-transversal set in G. In this paper, we first present a lower bound on τC(G) and characterize the extremal graphs achieving the lower bound for a connected (claw, K4)-free 4-regular graph G. Furthermore, we show that for any 2-connected (claw, K4)-free 4-regular graph G of order n, its clique-transversal number equals to ⌈n/3⌉.

To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S 9r (G) is finite, where S R (G):= sup{L(C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d C (x, y) ≤ d G (x, y) + R for every x, y ∈ C.

The aim of this paper is to prove the a.e. convergence of sequences of the Cesaro and Riesz means of the Walsh-Fourier series of d variable integrable functions. That is, let a = (a1, ...,ad): ℕ → ℕd (d ∈ ℙ) such that aj(n + 1) ≥ δ supk≤naj(k) (j = 1, ..., d, n ∈ ℕ) for some δ > 0 and a1(+∞) = ... = ad(+∞) = +∞. Then, for each integrable function f ∈ L1(Id), we have the a.e. relation for the Cesaro means limn→∞ σa(n)αf = f and for the Riesz means limn→∞ σa(n)α,γf = f for any 0 < αj ≤ 1 ≤ γj (j = 1, ..., d). A straightforward consequence of our result is the so-called cone restricted a.e. convergence of the multidimensional Cesaro and Riesz means of integrable functions, which was proved earlier by Weisz.

In this paper, we show that the nonorientable genus of C m + C n , the join of two cycles C m and C n , is equal to $\left\lceil {\tfrac{{(m - 2)(n - 2)}} {2}} \right\rceil $ if m = 3, n ≡ 1 (mod 2), or m ≥ 4, n ≥ 4, (m, n) ≠ (4, 4). We determine that the nonorientable genus of C 4 +C 4 is 3, and that the nonorientable genus of C 3 +C n is $\tfrac{n} {2} $ if n ≡ 0 (mod 2). Our results show that a minimum nonorientable genus embedding of the complete bipartite graph K m,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.

In this paper, we show that, for each p > 1, there are continuum many Borel equivalence relations between ℝωl1 and ℝωlp ordered by ≤B which are pairwise Borel incomparable.

A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V (G) − S is also adjacent to a vertex in V (G) − S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the study of total restrained bondage in graphs. The total restrained bondage number in a graph G with no isolated vertex, is the minimum cardinality of a subset of edges E such that G - E has no isolated vertex and the total restrained domination number of G - E is greater than the total restrained domination number of G. We obtain several properties, exact values and bounds for the total restrained bondage number of a graph.

Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term “twisting” and “2-switching” respectively. With the twisting operation, we give a pure geometrical proof of Whitney’s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs.