Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

Series-parallel chromatic hypergraphs

In this paper two-terminal series-parallel chromatic hypergraphs are introduced and for this class of hypergraphs it is shown that the chromatic polynomial can be computed with polynomial complexity. It is also proved that h-uniform multibridge hypergraphs θ(h;a₁,a₂,…,a_k) are chromatically unique for h≥3 if and only if h=3 and a₁=a₂=?=a_k=1, i.e., when they are sunflower hypergraphs having a core of cardinality 2 and all petals being singletons.     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.     

3-Factor-criticality in domination critical graphs

A graph G is said to be k-γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k-1 vertices. The structure of k-γ-critical graphs remains far from completely understood when k?3.A graph G is factor-critical if G-v has a perfect matching for every vertex v∈V(G) and is bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v∈V(G). More generally, a graph is said to be k-factor-critical if G-S has a perfect matching for every set S of k vertices in G. In three previous papers [N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs, Discrete Math. 272 (2003) 5-15; N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs. II. Utilitas Math. 70 (2006) 11-32], we explored the toughness of 3-γ-critical graphs and some of their matching properties. In particular, we obtained some properties which are sufficient for a 3-γ-critical graph to be factor-critical and, respectively, bicritical. In the present work, we obtain similar results for k-factor-critical graphs when k=3.     

Enumeration of bilaterally symmetric 3-noncrossing partitions

Schützenberger’s theorem for the ordinary RSK correspondence naturally extends to Chen et al.’s correspondence for matchings and partitions. Thus the counting of bilaterally symmetric k-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions.     

Using Lagrangians of hypergraphs to find non-jumping numbers(II)

Let r?2 be an integer. A real number α∈[0,1) is a jump for r if for any ε>0 and any integer m?r, any r-uniform graph with n>n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c=c(α)>0 does not depend on ε and m. A result of Erd?s, Stone and Simonovits implies that every α∈[0,1) is a jump for r=2. Erd?s asked whether the same is true for r?3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumping numbers for every r?3. However, there are a lot of unknowns on determining whether or not a number is a jump for r?3. In this paper, we find two infinite sequences of non-jumping numbers for r=4, and extend one of the results to every r?4. Our approach is still based on the approach developed by Frankl and Rödl.     

Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H. In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρ and σ on H we investigate when the hypergroups associated with the relations ρ∩σ, ρ∪σ and ρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.     

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.     

A group-based search for solutions of the n-queens problem

The n-queens problem is a well-known problem in mathematics, yet a full search for n-queens solutions has been tackled until now using only simple algorithms (with the exception of the Rivin-Zabih algorithm). In this article, we discuss optimizations that mainly rely on group actions on the set of n-queens solutions. Most of our arguments deal with the case of toroidal queens; at the end, the application to the regular n-queens problem is discussed, and also the Rivin-Zabih algorithm.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

The role of directional compactness in the existence and computation of contingent epiderivatives

This paper underlines the role of directional compactness in the scalarization of graphical derivatives of set-valued maps taking values in infinite-dimensional spaces. Two main theorems are given. The first one states the equivalence of contingent epiderivatives and τw-contingent epiderivatives for directionally compact maps. The second main result proves a variational characterization for the contingent epiderivative of stable and directionally compact maps taking values in general image spaces, extending known results in finite-dimensional and reflexive Banach spaces. The hypotheses given are minimal as is shown by means of several examples. Connections of these theorems with other results of the literature are also provided.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.     

A degree sum condition for graphs to be prism hamiltonian

Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ₃(G)≥n is prism hamiltonian. The degree sum condition “σ₃(G)≥n” is best possible.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

The H-covariant strong Picard groupoid

The notion of H-covariant strong Morita equivalence is introduced for ^*-algebras over C=R(i) with an ordered ring R which are equipped with a ^*-action of a Hopf ^*-algebra H. This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant ^*-Morita equivalence with its H-covariant ^*-Picard groupoid. We discuss various groupoid morphisms between the corresponding notions of the Picard groupoids. Moreover, we realize several Morita invariants in this context as arising from actions of the H-covariant strong Picard groupoid. Crossed products and their Morita theory are investigated using a groupoid morphism from the H-covariant strong Picard groupoid into the strong Picard groupoid of the crossed products.     

A new Bartholdi zeta function of a digraph II

We introduce a new type of the Bartholdi zeta function of a digraph D. Furthermore, we define a new type of the Bartholdi L-function of D, and give a determinant expression of it. We show that this L-function of D is equal to the L-function of D defined in [H. Mizuno, I. Sato, A new Bartholdi zeta function of a digraph, Linear Algebra Appl. 423 (2007) 498-511]. As a corollary, we obtain a decomposition formula for a new type of the Bartholdi zeta function of a group covering of D by new Bartholdi L-functions of D.