1-平面图的结构性质及其在无圈边染色上的应用

一个图称为是1-平面的如果它可以画在一个平面上使得它的每条边最多交叉另外一条边.本文描述了任意1-平面图中小于等于7度点之邻域的局部结构,解决了由Fabrici和Madaras提出的两个关于1-平面图图类中轻图存在性的问题,证明了每个最大度是△的1-平面图G是无圈列表max{2△-2,△+83}-边可选的.     

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.     

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.     

1-平面图及其子类的染色

如果图G可以嵌入在平面上,使得每条边最多被交叉1次,则称其为1-可平面图,该平面嵌入称为1-平面图.由于1-平面图G中的交叉点是图G的某两条边交叉产生的,故图G中的每个交叉点c都可以与图G中的四个顶点(即产生c的两条交叉边所关联的四个顶点)所构成的点集建立对应关系,称这个对应关系为θ.对于1-平面图G中任何两个不同的交叉点c_1与c_2(如果存在的话),如果|θ(c_1)∩θ(c_2)|≤1,则称图G是NIC-平面图;如果|θ(c_1)∩θ(c_2)|=0,即θ(c_1)∩θ(c_2)=?,则称图G是IC-平面图.如果图G可以嵌入在平面上,使得其所有顶点都分布在图G的外部面上,并且每条边最多被交叉一次,则称图G为外1-可平面图.满足上述条件的外1-可平面图的平面嵌入称为外1-平面图.现主要介绍关于以上四类图在染色方面的结果.     

1.

一个平面图G被称为1-树如果存在一个顶点u使得G-u是一个林.本文确定了所有1-树的边面全色数的精确上、下界,并且求出了2-连通且最大度至少为6平面图的边面全色数.     

最大度为6的平面图为第一类的一个新充分条件

本文证明了:若一个平面图G不含带弦的6-圈,则G是第一类的.这部分地证实了Vizing的关于平面图边染色的一个猜想.     

哈林图的弱边面染色

设G=(V,E,F)是一个无环的连通平面图,其中V表示点集,E表示边集,F表示面集.对于任意的两条相邻边e_1和e2,如果它们关联同一个面且在该面的边界上连续出现,那么称e_1和e2是面相邻的.图G是弱边面k-可染的是指存在一个映射π:EUF→{1,2,…,k},使得任意两个相关联的边和面,任意两个相邻的面,以及任意两条面相邻的边都染不同的颜色.平面图G的弱边面染色数是指G是弱边面k-可染的数k的最小值,用_(ef)(G)表示.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.本文我们给出此猜想的一个充分条件,即证明:哈林图是弱边面5-可染的,其中上界5是最好可能的.     

平面图上的团横贯数与独立数

设G为简单图,若G的点子集S与图中的每个团都有非空的交,则称S是图G的一个团横贯集,这里G的团是指图中的极大完全子图且至少包含两个点.图G的最小团横贯集所含点的数目称为G的团横贯数,记作τC(G).如果G的每条边至少包含在一个t阶完全子图中且τC(G)≤|V(G)|/t,则称G具有〈t〉一性质.提出了平面图分离4-团的概念.首先证明了最大度不超过5的平面图具有〈t〉-性质.其次,对任意平面图G,若它不含分离4-团且每条边都包含在一个4-团之中,得到了它的横贯数的上界和独立数的可达下界.     

第二大特征值小于$\frac{\sqrt{5}-1}{2}$的平面图

若能将图$G$画在一个平面上,使得任何两条边仅在顶点处相交,则称$G$是平面图.本文刻画了第二大特征值小于$\frac{\sqrt{5}-1}{2}$的所有无孤立点的平面图.     

不含5-圈的平面图的无圈边着色

图G的一个无圈边着色是一个正常的边着色且不含双色的圈.图G的无圈边色数是图G的无圈边着色中所用色数的最小者.本文用反证法得到了不含5-圈的平面图G的无圈边色数的一个上界.     

The structure of 1-planar graphs

A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an edge with both endvertices of degrees at most 20, and we present similar results concerning bigger structures in 1-planar graphs with additional constraints.     

The height of a 4-cycle in triangle-free 1-planar graphs with minimum degree 5

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. It is known that each 1-planar graph has a vertex of degree at most 7, and also either a vertex of degree at most 4 or a cycle of length at most 4. In the article, it is proven that each triangle-free 1-planar graph of degree less than 5 has a 4-cycle that consists of vertices of degree at most 8.     

List edge and list total coloring of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that each 1-planar graph with maximum degree Δ is (Δ+1)-edge-choosable and (Δ+2)- total-choosable if Δ≥16, and is Δ-edge-choosable and (Δ+1)-total-choosable if Δ≥21. The second conclusion confirms the list coloring conjecture for the class of 1-planar graphs with large maximum degree.     

Equitable Coloring of Three Classes of 1-planar Graphs

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent crossings or independent crossings, say NIC-planar graph or IC-planar graph, is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex or no common vertices, respectively. In this paper, we prove that each 1-planar graph, NIC-planar graph or IC-planar graph with maximum degree Δ at least 15, 13 or 12 has an equitable Δ-coloring, respectively. This verifies the well-known Chen-Lih-Wu Conjecture for three classes of 1-planar graphs and improves some known results.     

Drawing complete multipartite graphs on the plane with restrictions on crossings

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near independent crossings （say NIC-planar graph） is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex. The full characterization of NIC-planar complete and complete multipartite graphs is given in this paper.     

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.In this paper,we prove that every 1-planar graph G with maximum degree Δ(G)≥12 and girth at least five is totally(Δ(G)+1)-colorable.     

Acyclic edge coloring of triangle-free 1-planar graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ a(G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ a(G) ≤Δ(G) + 22, if G is a triangle-free 1-planar graph.     

Minimal non-1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G-e is 1-planar for every edge e of G. We prove that there are infinitely many minimal non-1-planar graphs (MN-graphs). It is known that every 6-vertex graph is 1-planar. We show that the graph K₇-K₃ is the unique 7-vertex MN-graph.