Topological graphTopological graph theory discusses, in most cases, graphs embedded in the plane (or other surfaces). For example, such plane graphs are sometimes regarded as the simplest town maps. Now, we consider a town having some pedestrian bridges, which cannot be realized by a plane ...
A graph is 1-planarif it can be drawn on the plane so that each edge is crossed by at most one other edge. Such a 1-planar drawing of G is called a 1-plane graph. Clearly, the family of 1-planar graphs is bigger than that of planar ones. The following Figure 2 is a 1-planar...
In this paper, we improved this result and showed that every 1-planar graph is odd 13-colorable. Introduction A proper k-coloring of a graph G is a mapping f:V(G)→[k] such that f(u)≠f(v) whenever uv∈E(G), where [k]={1,2,…,k}. An odd coloring of a graph G is a...
1-Planar GraphCombinatorial ProblemThe bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G....
In a book embedding the vertices of a graph are placed on the “spine” of a book and the edges are assigned to “pages”, so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such th...
For example, Fabrici and Madaras [2] showed that each 1-planar graph contains a vertex of degree at most 7, and proved that each 3-connected 1-planar graph contains an edge with both endvertices of degrees at most 20. Here the bound 20 is also sharp. As we know, every 3-connected...
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 Delta is (Delta+1)-edge-choosable and (Delta+2)-total-choosable if Delta
A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite $1$-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most $3n...
A simple graph G is said to be 1-planar if it can be drawn on the sphere S2(or the plane) so that each of its edges crosses at most one other edge at a point. The drawing is regarded as a continuous map f : G → S2 which may not be injective. To simplify our notation, we...
1、Planar Graphs,2,2020/9/4,College of Computer Science & Technology, BUPT,Planar Graphs 平面图,A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph: The clique on 4 nodes.,3,2020/9/4,College of Computer ...