The M-degree of an edge xy in a graph is the maximum of the degrees of x and y. The M-degree of a graph G is the minimum over M-degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4cycles has M-degree at most 8 and gave an example of such a graph with M-degree 3. This yields upper bounds on the game chroma...

There are planar class 2 graphs with maximum vertex-degree k, for each k ∈ {2, 3, 4, 5}. In 1965, Vizing proved that every planar graph with ∆ ≥ 8 is class 1. He conjectured that every planar graph with ∆ ≥ 6 is a class 1 graph. This conjecture is proved for ∆ = 7, and still open for ∆ = 6. Let k ≥ 2 and G be a k-critical planar graph. The average face-degree F (G) of G is 2 |F (G)| |E(G)|. Let...

We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G′ of G such that adding to G′ an extra edge of G results in a non-planar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new opt...

Steinberg asked whether every planar graph without 4 and 5 cycles is 3-colorable. Borodin, and independently Sanders and Zhao, showed that every planar graph without any cycle of length between 4 and 9 is 3-colorable. We improve this result by showing that every planar graph without any cycle of length 4, 5, 6, or 9 is 3-choosable. © 2005 Elsevier B.V. All rights reserved.

The problems under consideration here were united for the first time by the general term “The Three Color Problem” in Ore’s book [7], where the author was interested in the description of planar 3-colorable graphs. Our graph theoretic terminology is that of Harary [4], except that we use vertices and edges instead of points and lines, respectively. The graphs considered here are planar and have...

We study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs.

We study contact representations of non-planar graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We present a liner-time algorithm constructing a representation of a 3-connected planar graph, its dual, and the vertex-face incidence graph with 3D boxes. We then investigate contact re...

It is known that for any orientable surface Sg other than the sphere, there exists an optimal 1-planar graph which can be embedded on Sg as a triangulation. In this paper, we prove that for any orientable surface Sg with genus g ≥ 3 and any non-orientable surface Nk with genus k ≥ 6 (k = 7), there exists an optimal 1-planar graph which can be embedded on the surface as a quadrangulation. Furthe...

We construct an optimal linear time algorithm for the maximal planar subgraph problem: given a graph G, nd a planar subgraph G 0 of G such that adding to G 0 any edge of G not present in G 0 leads to a non-planar graph. Our solution is based on a dynamic graph search procedure and a fast data structure for on-line planarity testing of triconnected graphs. Our algorithm can be transformed into a...

A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but all are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to rst st-number the vertices). Our new algor...