The centralizing theorem for left normal groups of units in compact monoids

A group G of units in a monoid S is called left normal if sG ⊆ Gs for all s ε S. The centralizer Z of G in S is the set of all s ε S with sg=gs for all g ε G. Always l ε Z, and if S has a zero 0, then 0 ε Z. We show that in a compact connected monoid S with zero the centralizer Z of any left normal (closed) group G of units is connected. This work was supported by NSF.     

The universal group of homogeneous quotients

This paper constructs from the homogeneous quotients of an arbitrary semigroupS a universal group (G(S), γ) onS. If S is left reversible and cancellative, thenG(S) coincides with the embedding group of quotients of S due to Ore. If S is an inverse semigroup, G(S) coincides with the maximum group homomorphic image of S due to Munn. In these cases, γ coincides with the embedding and canonical homomorphism respectively ofS intoG(S). In general (G(S), γ) is equivalent to the universal group on S due to N. Bouleau. A universal group constructed from the set of Lambek ratios had earlier been exhibited by A.H. Clifford and G.B. Preston for cancellative semigroups satisfying the condition Z of Malcev. No previous construction has, however, emerged as a direct generalisation of both the work of Ore and Munn as does the present. Elementary properties of homogeneous quotients are employed to illuminate Bouleau's counter-example on why certain Malcev conditions are insufficient to guarantee the embeddability of a semigroup in a group.     

A counterexample to the bold conjecture

A pair of vertices (x,y) of a graph G is an o-critical pair if o(G + xy) > o(G), where G + xy denotes the graph obtained by adding the edge xy to G and o(H) is the clique number of H. The o-critical pairs are never edges in G. A maximal stable set S of G is called a forced color class of G if S meets every o-clique of G, and o-critical pairs within S form a connected graph. In 1993, G. Bacsó raised the following conjecture which implies the famous Strong Perfect Graph Conjecture: If G is a uniquely o-colorable perfect graph, then G has at least one forced color class. This conjecture is called the Bold Conjecture. Here we show a simple counterexample to it. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 165–168, 1997     

On barrier sets of star-factors

A barrier set of a graphG for a star-factor is a setS ofV(G) such thati(G – S) > k|S|, wherei(G – S) denotes the number of isolated vertices ofG – S. In this paper, we obtain some results on barrier sets.     

三角环的全可导点

设A是一个结合环,G∈A.G称为A的一个全可导点,如果每一个在G点可导的可加映射φ:A→A(即对任意的S,T∈A有φ(ST)=φ(S)T+Sφ(T)且ST=G)都是一个导子.本文证明了一类三角环上的每个非零元都是全可导点.作为此结果的推论得到:一类域上的三角矩阵环的每个非零元都是全可导点.     

THE NORMALITY OF CAYLEY GRAPHS OF SIMPLE GROUP A5 WITH SMALL VALENCIES

Let G be a finite group and S a subset of G not containing the identity element 1. We define the Cayley (di)graph X = Cay(G, S) of G with respect to S by V(X) = G,E(X) = {(g, sg) [ g ∈ G, s ∈ S}. A Cayley (di)graph X = Cay(G, S) is called normal if GR A = Aut(X). In this paper we prove that if S = {a, b, c} is a 3-generating subset of G = A5 not containing the identity 1, then X = Cay(G, S) is a normal Cayley digraph.     

On finite groups with a reducible permutability-graph

We characterize finite groups in which the permutability-graph has more than one connected component.Research partially supported by G.N.S.A.G.A. of C.N.R. and M.U.R.S.T. of Italy.     

Stratifications et ensemble de non-cohérence d'un espace analytique réel

Sans résuméL'auteur est associé au groupe G.N.S.A.G.A. du C.N.R.     

The geometrical structure in a generic generating space of a non-developable generalized ruled surface in the hyperbolic space Hm

A generalized ruled surface (G.R.S.) in H^m is generated by ¹ n-dimensional totally geodesic subspaces of H^m, which are called generators of the G.R.S. In this paper the basic results about points of striction, parameters of distribution and Riemann curvature at the points of a fixed generator of the G.R.S. are obtained, using a method which makes any representation of the G.R.S. in H^m superfluous.     

A characterization of discrete groups

The purpose of this article is to prove the following result. Let be a locally compact group, the Fourier algebra of and ~0$"> such that . Then is a discrete group .

     

Arcs in Minkowski planes

The aim of this paper is to study some properties of k-arcs in Minkowski planes focalizing the attention on problems of existence and completness.Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.In memoriam Giuseppe Tallini     

有关循环图C(n;{1,k})的独立数的一些结果

令G=(V(G),E(G))是一个简单有限无向图.如果V(G)的子集S中任意两个顶点均不相邻,则S是图G的一个独立集.顶点独立集大小的最大值,称为图G的独立数,记作α(G).本文研究了循环图C(n;{1,k})的独立数问题,并给出了当k=2,3,4,5时的准确值.     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

Derived semifield planes of odd order admitting a dihedral group of affine homologies

Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.Dedicated to A. Barlotti on the occasion of his 65. birthday.Research partially supported by G.N.S.A.G.A. (C.N.R.)     

Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ₄×ℤ₂ orG≅Q ₈×ℤ ₂ ^r (r⩾0) and that every finite group has a normal Cayley digraph, where Z_m is the cyclic group of orderm and Q₈ is the quaternion group of order 8. Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.     

自同构群是循环群被交换群扩张的有限群

设Ｃ是有限群，ＡｕｔＧ＝ＡＢ，，Ａ是交换群且每Ｓｙｌｏｗ子群的秩≤２，Ｂ是循环群，本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤２的交换群时，Ｇ循环。     

A note on a Heegaard diagram of S3

In this work, we study via crystallizations some properties of the counterexample to the Whitehead's conjecture and to the algorithm of Volodin-Kuznetsov-Fomenko, introduced by Ochiai. Research performed under the auspices of the G.N.S.A.G.A.-C.N.R., and within the Project “Topologia e geometria”, supported by M.U.R.S.T. of Italy     

La geometria asintotica delle foglie « eccezionali » delle foliazioni di Anosov

Résumé On considère les feuilletages d'Anosov de T₁S, avec S surface hyperbolique fermée, et on étudie la géométrie asymptotique des feuilles « exceptionnelles ».

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R. e col contributo del M.P.I. (fondi 60%).     

On a characterization of Dilworth truncations of combinatorial geometries

A combinatorial geometry being given, a Dilworth structure is defined to be a family of point subsets for which properties (1_d), (2_d), (3_d) in sect.2 hold. Let T^d(G) denote Dilworth truncation of a geometry G. It is possible to associa te with T^d(G) a Dilworth structure D(G) (see sect.2). It will be proved that a one-to-one and onto corresponden ce exists between Dilworth structures S of a connected geo metry K and those geometries G such that T^d(G)=K and D(G)=S.     

On Domination and Independence Numbers of Graphs

A set S of vertices of a graph G is dominating if each vertex x not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number, γ(G), is the order of the smallest dominating set in G. The independence number, α(G), is the order of the largest independent set in G. In this paper we characterize bipartite graphs and block graphs G for which γ(G) = α(G).