素数阶循环图和经典Ramsey数R（4，n）的三个新下界

研究了素数阶循环圈的基本性质，提出了寻求有效参数构造正则循环圈的新方法，得到了3个经典Ramsey数的新下界：R（4，17）≥164，R（4，18）≥182，R（4，22）≥282．这前2个结果填补了关于Ramsey数综述[2]的上下界表中的2个空白，第3个结果超过了目前已知的最好下界R（4，22）≥258，     

交叉数为2的笛卡尔积图

图G的交叉数,记作cr(G),是把G画在平面上的所有画法中边与边产生交叉的最小数目,它是拓扑图论中的一个热点问题。Kle?c和Petrillová刻画了当G₁为圈且cr(G₁G₂)-2时,因子图G₁和G₂满足的充要条件。在此基础上,本文研究当|V(G₁)|≥3且cr(G₁G₂)=2时,G₁和G₂应满足的充要条件。     

五阶图与路P_n的联图交叉数

利用Kleitman D J给出的完全二部图的的交叉数cr(_(5,n))=Z(5,n)的结果,分别得到了联图G_(12)∨P_n,G_(15)∨P_n,G_(18)∨P_n的交叉数.同时,给出了目前已知的所有五阶图与路的联图交叉数情况.     

关于一个特殊六阶图与路和圈的联图的交叉数

Garey和Johnson证明了确定图的交叉数问题是一个NP-难问题.目前,已确定交叉数的图类并不多.本文证明了一个特殊6阶图与n个孤立点,路P_n及圈C_n的联图的交叉数分别是cr(Q+nK_1)=Z(6,n)+n;cr(Q+P_n)=Z(6,n)+n+1及cr(Q+C_n)=Z(6,n)+n+3.     

五阶图与星图的笛卡尔积交叉数

In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr（G13 × Sn） = 4[n/2] [n-1/2]＋[n/2] . The result fills up the blank of the crossing numbers of Cartesian products of stars with all 5-vertex graphs presented by Marian Klesc.     

交叉数为2且因子图为路的笛卡尔积图

M.Kle??和J.Petrillová刻画了当G1为圈且cr (G1G2)=2时,因子图G1和G2所满足的充要条件.在此基础上,该文进一步刻画了在cr (G1G2)=2的前提下,当G1=P4,或者G1=P3且△(G2)=4时,因子图G2应满足的充要条件.     

循环图C(10,2)与路P_n的笛卡尔积的交叉数

证明了循环图C(10,2)与路P_n的笛卡尔积的交叉数是10n及循环图C(2m,2)的一点悬挂和两点悬挂的交叉数分别是m,2m.     

六阶图Q与星图S_n的积图交叉数

目前关于积图的交叉数的研究已经推广到六阶图与星图的积图.研究得到了一个特殊六阶图Q与n个孤立点nK_1的联图交叉数,然后通过收缩的方法,得到了Q与星图S_n的积图交叉数.     

循环图C(9,2)与路Pn的笛卡儿积的交叉数

C(m,2)表示由圈Cm(v1v2…vmv1)增加边vivi+2(i=1,…,m,i+2 (mod m))所得的循环图.C(m,2)的一点悬挂(两点悬挂)是增加一个顶点x(两个顶点x,y)和边xv(边xv,yv)的图,其中v∈V(C(m,2)).我们证明了9阶循环图C(9,2)与路Rn的笛卡儿积的交叉数是10n;C(2m-1,2)的一点悬挂和两点悬挂的交叉数分别是m,2m.     

On the Crossing Number of Circular Graphs

1.IntroductionInVLSIchipdesign,thetwo-layerroutingofgraphGplayanimportantrole.Thatis,theupperlayercanonlybeusedforverticalwiringandthesecondlaer,theloweroneonlyforhorizontalwiring.Soweconsidertheplanarprojectionofth.etwolayers,i.e.,consideratwodimensionalgridasapropergraphmodel.ThenoneobviousconditiontoberequiredisthateveryvertexofGhasitsdegreeatmost4.Withoutlossofgeneralitylweonlyconsider4regUlargraphs.IfGisaplanargraphwitha(G)S4,alinearalgorithmhadbeenprovidedforfindingarectilineajrrout…     

Characterizing Graphs with Crossing Number at Least 2

Our main result includes the following, slightly surprising, fact: a 4‐connected nonplanar graph G has crossing number at least 2 if and only if, for every pair of edges having no common incident vertex, there are vertex‐disjoint cycles in G with one containing e and the other containing f.     

Triple Crossing Numbers of Graphs

We introduce the triple crossing number,a variation of the crossing number,of a graph,which is the minimal number of crossing points in all drawings of the graph with only triple crossings.It is defined to be zero for planar graphs,and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings.In this paper,we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.     

On the Pseudolinear Crossing Number

A drawing of a graph is pseudolinear if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The pseudolinear crossing number of a graph G is the minimum number of pairwise crossings of edges in a pseudolinear drawing of G. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.     

$W_m$与$P_n$ 的笛卡儿积图的交叉数

Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn, for arbitrary m ≥ 3 and n≥ 1.     

覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

可定向曲面上两个地图的交叉数

In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.     

一类非凸的不可定向交叉数的序列

It is well known that finding the crossing number of a graph on nonplanar surfaces is very difficult.In this paper we study the crossing number of the circular graph C（10,4） on the projective plane and determine the nonorientable crossing number sequence of C（10,4）.On the basis of the result,we show that the nonorientable crossing number sequence of C（10,4） is not convex.