关于平面图（3，1）＊-可选性的一个注记

给出了平面图的一个结构性定理,并证明了每个没有5-圈,相邻三角形,相邻四边形的平面图是（3,1）＊-可选色的．     

外平面图的弱边面染色

假设G是一个平面图.如果e1和e2是G中两条相邻边且在关联的面的边界上连续出现,那么称e1和e2面相邻.图G的一个弱边面κ-染色是指存在映射π:E∪F→{1,…,κ},使得任意两个相邻面、两条面相邻的边以及两个相关联的边和面都染不同的颜色.若图G有一个弱边面κ-染色,则称G是弱边面κ-可染的.平面图G的弱边面色数是指G是弱边面κ-可染的正整数κ的最小值,记为χef(G).2016年,Fabrici等人猜想:每个无环且无割边的连通平面图是弱边面5-可染的.本文证明了外平面图满足此猜想,即:外平面图是弱边面5-可染的.     

可平面图的r-hued染色(英文)

令k0,r0是两个整数.图G的一个r-hued染色是一个正常k-染色?使得每个度为d(v)的顶点v相邻至少min{d(v),r}个不同的颜色.图G的r-hued色数是使得G存在r-hued染色的最小整数k,记为χ_r(G).文章证明了,若G为不含i-圈,4≤i≤9,的可平面图,则χ_r(G)≤r+5.这一结果意味着对于无4-9圈的可平面图,r-hued染色猜想成立.     

n≤12阶(k,l)-正则极大平面图

我们知道当图的顶点数n>12时不存在正则极大平面图.相关文献提出了(k,l)-正则极大平面图的概念,并讨论了(5,6)-正则极大平面图的存在性.在相关文献中,作者分别讨论了阶n>12的(k,l)-正则极大平面图的存在条件及构造方法.本文讨论了阶n(≤12)的(k,l)-正则极大平面图的存在性,除两种情况外,本文给出了阶n(≤12)的(k,l)-正则极大平面图的存在条件及其一种构造的例子.     

2-连通的平面图的边面染色

一个平面图G的边面色数χ_(ef)(G)是最小的颜色数,使得G中任意两条相邻的边、两个相邻的面、以及两个关联的边和面都染不同的颜色.本文证明了,若G是?≥16的2-连通平面图,则χ_(ef)(G)=?.这改进了已知结果:若G是?≥24的2-连通平面图,则χ_(ef)(G)=?.     

不含5-圈和6-圈的平面图的(2,1)-全标号

图G的(2,1)-全标号是对图G的顶点和边的一个标号分配,使得:(1)任意两个相邻顶点标号不同;(2)任意两条相邻边标号不同;(3)任意顶点与其相关联的边标号至少相差2.两个标号的最大差值称为跨度,图G的所有(2,1)-全标号的最小跨度称为(2,1)-全标号数,记为λ_2~T(G).本文证明了如果G是一个?=p+5的平面图,且G不包含5-圈和6-圈,那么λ_2~T(G)=2?-p,p=1,2,3.     

IC-平面图为第一类图的一个充分条件

图G的一个正常k-边染色是指一个映射Φ:E(G)→{1,2,…,k},使得任意两条相邻的边x,y∈E(G)满足Φ(x)≠Φ(y).使得G具有正常k-边染色的最小正整数k称为图G的边色数,记为χ'(G).著名Vizing定理证明每个简单图G的边色数χ'(G)要么等于最大度Δ(G)要么等于Δ(G)+1.这个定理将所有的图分成了两类:第一类图满足关系式χ'(G)=Δ(G),第二类图满足关系式χ'(G)=Δ(G)十1.本文主要讨论特殊1-平面图的正常边染色问题.1-平面图G是指G能够嵌入到平面上使得G的任意一条边最多被交叉一次.1-平面图G按照上述条件的一种画法称为G的一种1-平面嵌入.所以1-平面图中的每个交叉点w都是由两条边相交所得,从而每个交叉点w都对应着两条相交边,同时也对应着由这两条相交边的四个端点组成的集合ψ(w).如果1-平面图的一个1-平面嵌入中任意两个交叉点w和w'满足ψ(w)∩ψ(w')=Φ,那么称此1-平面图为IC-平面图.在本文中,通过观察分析Δ-临界图和不含相邻弦6-圈的IC-平面图的结构,应用权值转移方法证明了任何最大度为7且不含相邻弦6-圈的IC-平面图G是第一类图.     

平面图的理论与四色问题(Ⅳ)——三着色与四色问题的几何表征

§16 三色问题 三色问题之有助于四色问题者乃是极大平面图的三色问题。因为实际上四色问题只需研究那些非3-可着色的极大平面图。可喜的是这点已得到完满解决。然,一般平面图的3-可着色的判定确非那样容易。本节着重于后者。 命题16.1 极大平面图3-可着色,当且仅当所有节点的次皆偶数。 证明 由推论8.2的对偶形式和推论8.1即得。 定理16.1 任何平面图4-可着色,当且仅当非Euler极大平面图4-可着色。 证明 由于一个图是Euler图,当且仅当其节点的次皆偶。必要性是直接的。充分     

Δ(G)=5的2-连通外平面图的邻点可区别全染色

Smarandachely邻点可区别全染色是指相邻点的色集合互不包含的邻点可区别全染色，是对邻点可区别全染色条件的进一步加强。本文研究了平面图的Smarandachely邻点可区别全染色，即根据2-连通外平面图的结构特点，利用分析法、数学归纳法，刻画了最大度为5的2-连通外平面图的Smarandachely邻点可区别全色数。证明了：如果$G$是一个$\Delta （G）=5$的2-连通外平面图，则$\chi_{\rm sat}（G）\leqslant 9$。     

图论与中学数学竞赛(二)

六平面图平面图一个图G,如果能够把它画在平面上,且除端点外任意两条边均不相交则称G可以嵌入平面,如果图G可以嵌入平面,则称G为可平面图可平面图在平面上的一个嵌入称为一个平面图,如图17—a所示的图G是一个可平面图。图17-b所示的图G是G的一个平面嵌入即平面图。     

Every planar graph without 4-cycles adjacent to two triangles is DP-4-colorable

Wang and Lih (2002) conjectured that every planar graph without adjacent triangles is 4-choosable. In this paper, we prove that every planar graph without any 4-cycle adjacent to two triangles is DP-4-colorable, which improves the results of Lam et al. (1999), Cheng et al. (2016) and Kim and Yu [ arXiv:1709.09809v1].     

Planar graphs without 4‐cycles adjacent to 3‐cycles are list vertex 2‐arborable

It is known that not all planar graphs are 4‐choosable; neither all of them are vertex 2‐arborable. However, planar graphs without 4‐cycles and even those without 4‐cycles adjacent to 3‐cycles are known to be 4‐choosable. We extend this last result in terms of covering the vertices of a graph by induced subgraphs of variable degeneracy. In particular, we prove that every planar graph without 4‐cycles adjacent to 3‐cycles can be covered by two induced forests. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 234–240, 2009     

Les hypercartes planaires sont des arbres tres bien etiquetes

R. Cori and B. Vauquelin have constructed (cf[1]) a one to one correspondence from rooted planar maps onto rooted well-labeled trees (trees whose vertices are labeled with natural numbers that differ by at most one on adjacent vertices). This correspondence does not associate other families of planar maps (e.g. planar hypermaps,...) and easily definable families of trees. The main result of this paper (Theorem 1, Section II) is to construct a new one to one correspondence from rooted planar maps onto rooted well-labeled trees which also associates rooted planar hypermaps with n edge-ends (called ‘brin’ in French) and rooted very well-labeled trees (well labeled trees whose adjacent vertices have not the same label) with n edges. This last result is given in Section 3, Theorem 2.The coding of rooting very well-labeled trees by words extending Dyck's words (or parenthesis systems), allows their enumeration, hence the enumeration of rooted planar hypermaps. This side is the subject of a work in progress under B. Vauquelin.     

围长至少为4的平面图的邻点可区别边色数

图G的邻点可区别边染色是G的正常边染色,使得每一对相邻顶点有不同的颜色集合．G的邻点可区别边色数X＇a（G）是使得G有一个k-邻点可区别边染色的最小正整数七．本文证明了：若G是围长至少为4且最大度至少为6的平面图,则X＇a（G）≤△＋2．     

On well-covered triangulations: Part II

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97-108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations.     

Planar graphs without adjacent cycles of length at most seven are 3-colorable

We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3≤i≤j≤7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs.     

Group edge choosability of planar graphs without adjacent short cycles

In this paper,we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group(Δ(G)+1)-edge-choosable,and some planar graphs with large girth and maximum degree are groupΔ(G)-edge-choosable.     

Complexity results for minimum sum edge coloring

In the Minimum Sum Edge Coloring problem we have to assign positive integers to the edges of a graph such that adjacent edges receive different integers and the sum of the assigned numbers is minimal. We show that the problem is (a) NP-hard for planar bipartite graphs with maximum degree 3, (b) NP-hard for 3-regular planar graphs, (c) NP-hard for partial 2-trees, and (d) APX-hard for bipartite graphs.     

Every planar graph without adjacent cycles of length at most 8 is 3-choosable

DP-coloring as a generalization of list coloring was introduced by Dvořák and Postle in 2017, who proved that every planar graph without cycles from 4 to 8 is 3-choosable, which was conjectured by Borodin et al. in 2007. In this paper, we prove that every planar graph without adjacent cycles of length at most 8 is 3-choosable, which extends this result of Dvořák and Postle.