实二次代数整数环上的单位格的类数

用邻格方法及Siegelmass公式证明了实二次代数域Q(√d)上单位格种gen(In)n≥4)的类数h(In)=3当且仅当Q(√3):n=4;Q(√5);n=6;Q(√3);n=4;Q(√17):n=4.     

3,4跳图的平面性

For a graph G of size ε≥1 and its edge-induced subgraphs H1 and H2 of size r(1≤r≤ε), H1 is said to be obtained from H2 by an edge jump if there exist four distinct vertices u,v,w and x in G such that (u,v)∈E(H2), (w,x)∈ E(G)-E(H2) and H1=H2-(u,v)+(w,x). In this article, the r-jump graphs (r≥3) are discussed. A graph H is said to be an r-jump graph of G if its vertices correspond to the edge induced graph of size r in G and two vertices are adjacent if and only if one of the two corresponding subgraphs can be obtained from the other by an edge jump. For k≥2, the k-th iterated r-jump graph Jrk(G) is defined as Jr(Jrk-1(G)), where Jr1(G)=Jr(G).An infinite sequence{Gi} of graphs is planar if every graph Gi is planar. It is shown that there does not exist a graph G for which the sequence {J3k(G)} is planar, where k is any positive integer. Meanwhile,lim gen(J3k(G))=∞,where gen(G) denotes the genus of a graph G, if the sequencek→∞J3k(G) is defined for every positive integer k. As for the 4-jump gra     

SOME CLASSES OF UPPER EMBEDDABLE GRAPHS

1IntroductionSincetheintroductoryinvestigationofmaximumgenusbyNordhaus,Stewart,andWhite[3],theupperembeddabilityofgraphshasreceivedgreatemphasissofar.AlthoughLiu[16]andNebesky[9]haverespectivelyprovideddifferentnecessaryandsufficientconditionsontheupperembeddabilityofgraphs,weknowlessaboutwhatclassesofgraphsareupperembeddable.Jaeger,PayanandXuong[10]provedthateverygraphobtainedbyconnecting(withanynumberofedges)twovertex-disjointupperembeddablegraphswithevenBettinumberisupperembeddable.Skov…     

Notes on braidzel surfaces for links

As a generalization of pretzel surfaces, L. Rudolph has introduced a notion of braidzel surfaces in his study of the quasipositivity for pretzel surfaces. In this paper, we show that any oriented link has a braidzel surface. We also introduce a new geometric numerical invariant of links with respect to their braidzel surface and study relationships among them and other ``genus' for links.

     

LOWER BOUND NUMBER OF IRREDUCIBLE GRAPHS ON SURMCES

1IntroductionThesurfaceisacompact2-manifold.AnorientablesurfaCeofgenuskishomeonlorphictoaspherewithkhandles,whichwedenoteitbySk.Anon-orientablesurfaceofgenuskishomeomorphictoaspherewithkcrosscaps,whichwedenoteitbySa.AnembeddingMonsurfaceSgmeanstheunderlyinggraphofMisdrawnonSOsuchthatnoedgesintersecteachotherandeachfaceishomeomorphictothedisc[5],wedenoteitbyMCS,(orforS.).Irreduciblegraphscharacterizethosegraphswhichcannotbeembeddedonasurfacebutdeletinganyedgeofthisgraph,theresultantgraphc…     

Genus-One Correction to Multicut Matrix Model Solutions

We calculate genus-one corrections to the Hermitian one-matrix model solution with an arbitrary number of cuts based on the loop equation, confirming the answer previously obtained from algebro-geometric considerations and generalizing it to the case of arbitrary potentials.     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

Curves of maximum genus in range A and stick-figures

In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than , with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Fløystad.

     

广义置换图的最大亏格(英文)

In this paper we prove that the generalized permutation graph G(n,k) is upper embeddable if it has at most two odd subcycles,and that the maximum genus of G(n,k) is more than[β(G(n,k))/3]in most cases.     

The genus polynomials of cross-ladder digraphs in orientable surfaces

Some results about the genus distributions of graphs are known,but little is known about those of digraphs.In this paper,the method of joint trees initiated by Liu is generalized to compute the embedding genus distributions of digraphs in orientable surfaces.The genus polynomials for a new kind of 4-regular digraphs called the cross-ladders in orientable surfaces are obtained.These results are close to solving the third problem given by Bonnington et al.