On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.     

Finite groups with minimal 1-PIM

Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).     

A New Bound on the Total Domination Subdivision Number

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3,

Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.     

Orthocomplemented lattices in ordered groups

On a partially ordered set G the orthogonality relation is defined by incomparability and is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space of the lattice has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We will also prove: if G is an ordered group with a positive cone P, then the lattice has the covering property iff , where g is an element of G and M is the intersection of all maximal subgroups contained in . Received August 1, 2006; accepted in final form May 29, 2007.     

The Unbalance of Set Systems

The unbalance of an intersecting family is defined as , where is the maximum degree of i.e. the maximum of over all vertices x. We show that the unbalance of a k-uniform intersecting family is at most when n ≥ 6k ³ and we determine all families achieving this bound.     

Linear Chromatic Bounds for a Subfamily of 3K 1-free Graphs

For an arbitrary class of graphs , there may not exist a function f such that , for every . When such a function exists, it is called a χ-binding function for . The problem of finding an optimal χ-binding function for the class of 3K ₁-free graphs is open. In this paper, we obtain linear χ-binding function for the class of {3K ₁, H}-free graphs, where H is one of the following graphs: , House graph and Kite graph. We first describe structures of these graphs and then derive χ-binding functions.     

Generalized Thrackles and Geometric Graphs in $${\mathbb{R}}^3$$ with No Pair of Strongly Avoiding Edges

We define the notion of a geometric graph in . This is a graph drawn in with its vertices drawn as points and its edges as straight line segments connecting corresponding points. We call two disjoint edges of G strongly avoiding if there exists an orthogonal projection of to a two dimensional plane H such that the projections of the two edges on H are contained in two different rays, respectively, with a common apex that create a non-acute angle. We show that a geometric graph on n vertices in with no pair of strongly avoiding edges has at most 2n − 2 edges. As a consequence we get a new proof to Vázsonyi’s conjecture about the maximum number of diameters in a set of n points in . This research was supported by THE ISRELI SCIENCE FOUNDATION (grant No. 938/06).     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any , let , and , and and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.      

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S ³ and Λ(G) is the union of all complex projective lines in which are tangent to S ³ at a point in L(G).     

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

Fibred multilinks and singularities $${f \overline g}$$

In this article we extend Milnor’s fibration theorem to the case of functions of the form with f, g holomorphic, defined on a complex analytic (possibly singular) germ (X, 0). We further refine this fibration theorem by looking not only at the link of , but also at its multi-link structure, which is more subtle. We mostly focus on the case when X has complex dimension two. Our main result (Theorem 4.4) gives in this case the equivalence of the following three statements:

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra L ^∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ^∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ^∞(G) by to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to . We study the corresponding space of “σ-harmonic operators”, i.e., fixed points in under the action of σ. We show, under mild conditions on either σ or G, that is in fact a von Neumann subalgebra of . Our investigation of relies, in particular, on a notion of support for an arbitrary operator in that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to via ideals in , where denotes the trace class operators on L ²(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis. M. Neufang was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach. V. Runde was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach.     

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A _C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .      

On $${\mathcal{F}}$$ -supplemented subgroups of finite groups

Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H _G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).     

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

On the spectrum of lamplighter groups and percolation clusters

Let be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let be a finite group and the lamplighter group (wreath product) over with group of “lamps” . We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on X with parameter . The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilities on the percolation cluster. In particular, if the clusters of percolation with parameter are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Żuk, resp. Dicks and Schick regarding the case when is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter is always related with the Plancherel measure of a convolution operator by a signed measure on , where or another suitable group. M. Neuhauser’s research supported by the Marie-Curie Excellence Grant MEXT-CT-2004-517154. The research of W. Woess was partially supported by Austrian Science Fund (FWF) P18703-N18.