On unknotting numbers and four-dimensional clasp numbers of links

In this paper, we estimate the unknotting number and the four-dimensional clasp number of a link, considering the greatest euler characteristic for an oriented two-manifold in the four-ball bounded by the link. Combining with a result due to Rudolph, we prove that an inequality stronger than the Bennequin unknotting inequality actually holds for any link diagram. As an application we show the equality conjectured by Boileau and Weber for a closed positive braid diagram.

     

The incompatibility of crossing number and bridge number for knot diagrams

We study methods for computing the bridge number of a knot from a knot diagram. We prove equivalence between a geometric and a combinatorial definition of the bridge number of a knot diagram. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.     

Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T(3,l), , and T(4,5). Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.     

Concordance crosscap number of a knot

We define the concordance crosscap number _c(K) of a knot K asthe minimum crosscap number among all the knots concordant toK. The four-dimensional crosscap number *(K) is the minimumfirst Betti number of non-orientable surfaces smoothly embeddedin the four-dimensional ball, bounding the knot K. Clearly,*(K) _c(K). We construct two infinite sequences of knots forwhich *(K) < _c(K). In particular, the knot 7₄ is one of theexamples.     

The lower bound on independence number

Let G be a graph with degree sequence ( d_v). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least , where f_m+1( x) is a function greater than for x> 0. For a weighted graph G = ( V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least where w_v is the weight of v.     

Geometric indices and the Alexander polynomial of a knot

It is well-known that any Laurent polynomial satisfying and is the Alexander polynomial of a knot in . We show that can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

     

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p≥2, a set S⊆V(G) is a geodetic set of G if each vertex v∈V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for and determine the same for some special classes of graphs. For positive integers r,d and n≥d+1 with r≤d≤2r, there exists a connected graph G with , and . Also, for any positive integers 2≤a<b≤c, there exists a connected graph G such that g(G)=a, g_c(G)=b and . A subset T of a g_c-set S is called a forcing subset for S if S is the unique g_c-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected geodetic number of S, denoted by f_c(S), is the cardinality of a minimum forcing subset of S. The forcing connected geodetic number of G, denoted by f_c(G), is f_c(G)=min{f_c(S)}, where the minimum is taken over all g_c-sets S in G. It is shown that for every pair a,b of integers with 0≤a≤b−4, there exists a connected graph G such that f_c(G)=a and g_c(G)=b.     

Class number 5, 6 and 7

We outline the determination of all imaginary quadratic fields with class number 5, 6 or 7.

     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

On thek dimensional Piatetski-Shapiro prime number theorem

Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x ₁ c ₁,⋯,c _k ) denote the number of primesp withp⩽x, , where 1<c ₁<⋯<c _k are fixed constants. It is proved that π(x;c ₁,⋯,c _k ) has an asymptotic formula ifc ₁ ⁻¹ +⋯+c _k ⁻¹ >k−k/(4k ²+2). Project supported by the National Natural Science Foundation of China (Grant No. 19801021) and the Natural Science Foundation of Shandong Province (Grant No.Q98A02110).     

The number of knot group representations is not a Vassiliev invariant

For a finite group and a knot in the -sphere, let be the number of representations of the knot group into . In answer to a question of D.Altschuler we show that is either constant or not of finite type. Moreover, is constant if and only if is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant is bounded by some function of the braid index, the genus, or the unknotting number, then is either constant or not of finite type.

     

On the triple points of singular maps

The number of triple points (mod 2) of a self-transverse immersion of a closed 2n-manifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9].? If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer- and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds. Received: October 12, 2001     

超图的Alcuin数与其横贯数的关系

1000多年前, 英国著名学者Alcuin曾提出过一个古老的渡河问题, 即狼、羊和卷心菜的渡河问题. 最近, Prisner和Csorba等考虑了一般``冲突图"上的渡河问题. 将这一问题推广到超图$H=(V,\mathcal{E})$\,上, 考虑一类情况更一般的运输计划问题. 现在监管者 欲运输超图中的所有点\,(代表``items")\,渡河, 这里$V$的点子 形成超边 当且仅当这些点代表的``items"在无人监管的情况下不能留在一起. 超图$H$的Alcuin数是指超图$H$具有可行运输方案\,(即把$V$的点代表的``items" 全部运到河对岸)\,时船的最小容量. 给出了 $r$-一致完全二部超图和它的伴随超图, 以及$r$-一致超图的Alcuin数, 同时证明了判断$r$-一致超图是否为小船图是NP 困难的.     

The ratio of the irredundance number and the domination number for block-cactus graphs

Let γ(G) and ir(G) denote the domination number and the irredundance number of a graph G, respectively. Allan and Laskar [Proc. 9th Southeast Conf. on Combin., Graph Theory & Comp. (1978) 43–56] and Bollobás and Cockayne [J. Graph Theory (1979) 241–249] proved independently that γ(G) < 2ir(G) for any graph G. For a tree T, Damaschke [Discrete Math. (1991) 101–104] obtained the sharper estimation 2γ(T) < 3ir(T). Extending Damaschke's result, Volkmann [Discrete Math. (1998) 221–228] proved that 2γ(G) ≤ 3ir(G) for any block graph G and for any graph G with cyclomatic number μ(G) ≤ 2. Volkmann also conjectured that 5γ(G) < 8ir(G) for any cactus graph. In this article we show that if G is a block-cactus graph having π(G) induced cycles of length 2 (mod 4), then γ(G)(5π(G) + 4) ≤ ir(G)(8π(G) + 6). This result implies the inequality 5γ(G) < 8ir(G) for a block-cactus graph G, thus proving the above conjecture. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 139–149, 1998     

On the doubly connected domination number of a graph

For a given connected graph G = (V, E), a set is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.     

Computing all power integral bases in orders of totally real cyclic sextic number fields

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.