A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete graph. Then it is shown that for the theorem to be false there must exist a complete planar graph of h edges such that h > 4. Finally the theorem is proved to be ...

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs...

In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of H(G) is established for the class of locally planar graphs on arbitrary surfaces.

In this paper the new coloring of planar, VEF-coloring, will be introduced. A VEF coloring of a simple planar graph G is a proper coloring of all elements, including vertices, edges and faces of G. We will give two conjectures for the upper bound of VEF and VEF-list coloring of a simple planar graph. However, we will prove these conjectures for planar graphs with a maximum degree of at least 12...

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number ofK2k+1,q , for k 2.We prove tight bounds for complete graphs.We also study the rectiline...

In this paper, we extend the Grr otzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of H(G) is established for the class of locally planar graphs on arbitrary surfaces.

A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing. In a recent paper, w...

We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject to edge insertions and deletions and that allows queries that test whether the graph is currently planar, or whether a potentia...

It is shown that a planar graph can be partitioned into three linear forests. The sharpness of the result is also considered. In 1969, Chartrand and Kronk [2] showed that the vertex arboricity of a planar graph is at most 3. In other words, the vertex set of a planar graph can be partitioned into three sets each inducing a forest. In this paper we present an improvement on this result: that the...

In this paper we investigate the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is deened and studied. All subgraphs of a graph G, which are subdivisions of K 5 or K 3;3 , turn out to deene facets of this polytope. We also present computational experience with a br...