关于两类平面图及相关图的L(d1,d2)-标号问题

图G的L(2,1)标号是一个从顶点集V(G)到非负整数集的函数f(x),使得若d(x,y)=1,则|f(x)-f(y)|≥(2;若d(x,y)=2,则|f(x)-f(y)|≥1.图G的L(2,1)标号数λ(G)是使得G有max{f(v)V∈V(G)}=k的L(2,1)标号中的最小数k.Griggs和Yeh猜想对最大度为△的一般图G,有λ(G)≤△2.本文将L(2,1)-标号推广到L(d1,d2)-标号,并得出了平面三角剖分图、立体四面体剖分图、平面近四边形剖分图的L(d1,d2)-标号的上界,作为推论,本文证明了对上述几类图,有上述猜想成立.     

关于几类图的L(3,2,1)-标号问题

图G的L(2，1)-标号是一个从顶点集V(G)到非负整数集的函数f(x)，使得若d(z，y)=1则|f(x)-f(y)|≥2；若d(x，y)=2，则|f(x)-f(y)|≥1。图G的L(2，1)-标号数是λ(G)使得G有的max|f(v)：v∈V(G)|=κ的L(2，1)-标号中的最小数κ。本将L(2，1)-标号问题推广到更一般的情形即L(3，2，1)标号问题，并得到了平面三角剖分图、立体四面体剖分图的λ3(G)的上界。     

图的L(2,1)标号与移动通讯频率分配问题

图G的L（2，1）标号是一个从顶点集V（G）到非负整数集的函数f(x)，使得若d(x,y)=1，则|f(x)-f(y)|≥2；若d(x,y)=2，则|f(x)-f(y)|≥1。移动通讯频率分配问题可以转化为图的L（2，1）标号问题。本文首先给出平面格子图的L（2，1）标号，然后通过平面格子图及相关图的L（2，1）标号得到平面近正六边形剖分图的L（2，1）面标号，从而解决了移动通讯的频率分配问题。     

关于几类图的L(2,1)标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) ,使得若d(x ,y) =1 ,则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y) ｜≥ 1 .图G的L( 2 ,1 ) 标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .Griggs和Yeh猜想对最大度为Δ的一般图G ,有λ(G) ≤Δ2 .本文给出了Kneser图 ,Mycieklski图 ,Descartes图 ,Halin图的λ值的上界 ,并证明了上述猜想对以上几类图成立     

竖梯的局部替换图的L(d,1,1)-标号

图的L(d,1,1)-标号定义为顶点集V(G)到非负整数集的映射f,且当d(u,v)=1时,均有|f(u)-f(v)|≥d,当d(u,v)=2,3时,均有|f(u)-f(v)|≥1.不妨设0为最小标号,则称图G的所有L(d,1,1)-标号中的最大跨度max{f(v):v∈V(G)}的最小数为图的L(d,1,1)-标号数,记为λ_d(G).基本给出了竖梯的局部替换图的L(d,1,1)-标号数的确切值或界.     

Frequency Assignment through Combinatorial Optimization Approach

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|(?)2 if d(x, y)=1 and |f(x)-f(y)|(?)1 if d(x,y)=2. The L(2,1)-labeling numberλ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v) : v∈V(G)}=k. We study the L(3,2,1)-labeling which is a generalization of the L(2,1)-labeling on the graph formed by the (Cartesian) product and composition of 3 graphs and derive the upper bounds ofλs(G) of the graph.     

两类图的L(2.1)-标号数

图G的一个L(2.1)-标号是从顶点集V(G)到非负整数的一个函数f,使得若d(u,v)=1时,有|f(u)-f(v)|≥2;若d(u,v)=2时,有|f(u)-f(v)|≥1.图G的L(2.1)-标号数λ(G)是G的所有L(2.1)-标号下的跨度max{f(v):v∈V(G)}的最小数.图Fn+1*为扇图的路上每个顶点增加一个悬挂边得到的图.图Hn为轮图的圈上每个顶点增加一个悬挂边得到的图.本文确定了图Fn+1*与Hn的L(2.1)-标号数.     

关于图的L(3,2,1)-标号问题

图 G的 L (2 ,1) -标号是一个从顶点集 V(G)到非负整数集的函数 f (x) ,使得若 d(x,y) =1,则 | f (x)- f (y) |≥ 2 :若 d(x ,y) =2 ,则 | f (x) - f (y) |≥ 1.图 G的 L (2 ,1) -标号数λ(G)是使得 G有 max{ f (v) :v∈ V(G) } =k的 L(2 ,1) -标号中的最小数 k.本文将 L(2 ,1) -标号问题推广到更一般的情形即 L(3,2 ,1) -标号问题 ,并得出了细分图、Descartes图的 λ3 (G)的上界 .     

点接手镯图的L(1,1,1)-标号

图的L(1,1,1)-标号定义为顶点集V(G)到非负整数集的映射f,且当d(u,v)=1,2,3时,均有|f(u)-f(v)|≥1.不妨设0为最小标号,则称图G的所有L(1,1,1)-标号中的最大跨度f(v)的最小数为图的L(1,1,1)-标号数,记为λ(G).基本给出了点接手镯图的L(1,1,1)-标号数的确切值.     

关于图的距离标号问题

图G的L（2，1）-标号是一个从顶点V（G）集到非负整数集的函数f（x），使得若d（x，y）：1，则｜f（x）-f（y）｜≥2；若d（x，y）=2，则｜f（x）-f（y）｜≥1。图G的L（2，1）-标号数A（G）是使得G有max{f（v）：v∈V（G）}=k的L（2，1）-标号中的最小数愚。本文将L（2，1）-标号问题推广到更一般的情形即L（d1，d2，d3）-标号问题，并得出了复合图的λd1，d2，d3（G）的上界。     

A Metric of Constant Curvature on Polycycles

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions:(1) each internal face is an r-gon, where r ≥ 3;(2) the degree of each internal vertex is q, where q ≥ 3, and the degree of each boundary vertex is at most q and at least 2.Then it also possesses the following third property:(3) the vertices, the edges, and the internal faces form a cell complex.Simple examples show that conditions (1) and (2) are independent even provided condition (3) is satisfied. These are the defining conditions for an (r, q)-polycycle.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 223–233.Original Russian Text Copyright © 2005 by M. Deza, M. I. Shtogrin.     

Saturation numbers for families of graph subdivisions

For a family of graphs, a graph G is ‐saturated if G contains no member of as a subgraph, but for any edge in , contains some member of as a subgraph. The minimum number of edges in an ‐saturated graph of order n is denoted . A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let denote the family of H‐subdivisions, including H itself. In this paper, we study when H is one of or , obtaining several exact results and bounds. In particular, we determine exactly for and show for n sufficiently large that there exists a constant such that . For we show that will suffice, and that this can be improved slightly depending on the value of . We also give an upper bound on for all t and show that . This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge‐maximal graphs without a K₅‐minor have at least edges.     

Degree Condition for Subdivisions of Unicyclic Graphs

If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.     

Hitting times for random walks on subdivision and triangulation graphs

Let G be a connected graph. The subdivision graph of G, denoted by S(G), is the graph obtained from G by inserting a new vertex into every edge of G. The triangulation graph of G, denoted by R(G), is the graph obtained from G by adding, for each edge uv, a new vertex whose neighbours are u and v. In this paper, we first provide complete information for the eigenvalues and eigenvectors of the probability transition matrix of a random walk on S(G) (res. R(G)) in terms of those of G. Then we give an explicit formula for the expected hitting time between any two vertices of S(G) (res. R(G)) in terms of those of G. Finally, as applications, we show that, the relations between the resistance distances, the number of spanning trees and the multiplicative degree-Kirchhoff index of S(G) (res. R(G)) and G can all be deduced from our results directly.     

简单完整正则平面图

对简单完整正则平面图的特性和结构进行了分析和讨论 ,找出了简单完整正则平面图的可能的种类 .此外 ,对各种简单完整正则平面图的色数进行了求解 ,并用不同的方法给出了各个简单完整正则平面图的作色方案 .     

平面对偶图哈密顿性的一个充分条件

提出了平面单图的对偶图是哈密顿图的一个充分条件     

The spectrum of infinite regular line graphs

Let be an infinite -regular graph and its line graph. We consider discrete Laplacians on and , and show the exact relation between the spectrum of and that of . Our method is also applicable to -semiregular graphs, subdivision graphs and para-line graphs.

     

无8-,9-和10-圈的平面图的3-可选择性

寻找平面图是3-或者4-可选择的充分条件是图的染色理论中一个重要研究课题,本文研究了围长至少是4的特殊平面图的选择数,通过权转移的方法证明了每个围长至少是4且不合8-圈,9-圈和10-圈的平面图是3-可选择的．     

Edge‐face coloring of plane graphs with maximum degree nine

An edge‐face coloring of a plane graph with edge set E and face set F is a coloring of the elements of E∪F so that adjacent or incident elements receive different colors. Borodin [Discrete Math 128(1–3):21–33, 1994] proved that every plane graph of maximum degree Δ?10 can be edge‐face colored with Δ + 1 colors. We extend Borodin's result to the case where Δ = 9. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:332‐346, 2011