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20
The FourColour Theorem
, 1997
"... The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken’s in several respects. ..."
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Cited by 163 (16 self)
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The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken’s in several respects.
A New Proof Of The FourColour Theorem
 ELECTRON. RES. ANNOUNCE. AMER. MATH SOC
, 1996
"... The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announce another proof, still using a computer, but simpler than Appel and Haken's in several respects. ..."
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Cited by 38 (0 self)
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The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announce another proof, still using a computer, but simpler than Appel and Haken's in several respects.
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 32 (14 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Counting Triangulations of Planar Point Sets
"... We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43^n for the problem. More ..."
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Cited by 21 (8 self)
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We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43^n for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossingfree) straightline graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O ∗ (239.4 n)), spanning cycles (O ∗ (70.21 n)), spanning trees (160 n), and cyclefree graphs (O ∗ (202.5 n)).
Random triangulations of planar points sets
"... Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of ..."
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Cited by 19 (8 self)
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Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of S. We are primarily interested in the degree sequences of such random triangulations.
Counting plane graphs: crossgraph charging schemes
 Proc. 20th International Symposium on Graph Drawing (GD 2012), LNCS
"... Abstract. We study crossgraph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in ..."
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Cited by 6 (3 self)
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Abstract. We study crossgraph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of 1 O ∗ ( 187.53 N) for the maximum number of crossingfree straightedge graphs that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound 207.85 N in Hoffmann et al. [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and biconnected plane graphs), and obtain various bounds on expected vertexdegrees in graphs that are uniformly chosen from the set of all crossingfree straightedge graphs that can be embedded over a specific point set. We then show how to apply the crossgraph chargingscheme method for graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwisecrossing edges (more commonly known as kquasiplanar graphs). For k = 3 and k = 4, we prove that, for any set S of N points in the plane, the number of graphs that have a straightedge kquasiplanar embedding over S is only exponential in N. 1
The 5 Colour Theorem in Isabelle/Isar
 THEOREM PROVING IN HIGHER ORDER LOGICS, VOLUME 2410 OF LNCS
, 2002
"... Based on an inductive definition of triangulations, a theory of undirected planar graphs is developed in Isabelle/HOL. The proof of the 5 colour theorem is discussed in some detail, emphasizing the readability of the computer assisted proofs. ..."
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Based on an inductive definition of triangulations, a theory of undirected planar graphs is developed in Isabelle/HOL. The proof of the 5 colour theorem is discussed in some detail, emphasizing the readability of the computer assisted proofs.
Face Labeling Of Maximal Planar Graphs
, 2011
"... The labeling problem considered in this paper is called facelabeling of the maximal planar or triangular planar graphs in connection with the notion of the consistency. Several triangular planar and maximal planar graphs such as the wheels, the fans etc. are considered. ..."
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The labeling problem considered in this paper is called facelabeling of the maximal planar or triangular planar graphs in connection with the notion of the consistency. Several triangular planar and maximal planar graphs such as the wheels, the fans etc. are considered.