The covering graph of a lattice which is contained in an ordered set with a planar covering graph is itself planar. AMS subject classifications (1980). 06A10

A connected graph with a cyclomatic number k is said to be k-cyclic graph. We obtain the formula for of labelled non-planar pentacyclic graphs given vertices, and find asymptotics planar tetracyclic n vertices as → ∞. prove that under uniform probability distribution on set consideration, asymptotically equal 1089/1105, labeled 1591/1675.

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H ' such that H has a homomorphism to H' . If each graph in a class ~ has a homomorphism to the same H' , then H ' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs...

A touching triangle graph representation (TTG) of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the endvertices of e. We call Γ a proper TTG if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input graph. In this pa...

The main result of the papzer is that any planar graph with odd girth at least 10k À 7 has a homomorphism to the Kneser graph G 2k‡1 k , i.e. each vertex can be colored with k colors from the set f1; 2;. .. ; 2k ‡ 1g so that adjacent vertices have no colors in common. Thus, for example, if the odd girth of a planar graph is at least 13, then the graph has a homomorphism to G 5 2 , also known as...

A graph G is k-choosable if for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem o...

A plane graph is a planar graph with a fixed embedding in the plane. In a rectangular drawing of a plane graph, each vertex is drawn as a point, each edge is drawn as a horizontal or vertical line segment, and each face is drawn as a rectangle. A planar graph is said to have a rectangular drawing if at least one of its plane embeddings has a rectangular drawing. In this paper we give a linear-t...

Given a planar graph G the planar biconnectivity augmen tation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected Given a nonplanar and biconnected graph the maximum planar biconnected subgraph problem consists of removing the minimum number of edges so that planarity is achieved and biconnectivity is maintained Both problems are imp...

An (F , Fd)-partition of a graph is a vertex-partition into two sets F and Fd such that the graph induced by F is a forest and the one induced by Fd is a forest with maximum degree at most d. We prove that every triangle-free planar graph admits an (F , F5)-partition. Moreover we show that if for some integer d there exists a trianglefree planar graph that does not admit an (F , Fd)-partition, ...

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.