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48
An update on the fourcolor theorem
 Notices of the AMS
, 1998
"... very planar map of connected countries can be colored using four colors in such a way that countries with a common boundary segment (not just a point) receive different colors. It is amazing that such a simply stated result resisted proof for one and a quarter centuries, and even today it is not ye ..."
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very planar map of connected countries can be colored using four colors in such a way that countries with a common boundary segment (not just a point) receive different colors. It is amazing that such a simply stated result resisted proof for one and a quarter centuries, and even today it is not yet fully understood. In this article I concentrate on recent developments: equivalent formulations, a new proof, and progress on some generalizations. Brief History The FourColor Problem dates back to 1852 when Francis Guthrie, while trying to color the map of the counties of England, noticed that four colors sufficed. He asked his brother Frederick if it was true that any map can be colored using four colors in such a way that adjacent regions (i.e., those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is by Cayley in 1878. A year later the first “proof ” by Kempe appeared; its incorrectness was pointed out by Heawood eleven years later. Another failed proof was published by Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe proved the fivecolor theorem (Theorem 2 below) and discovered what became known as Kempe chains, and Tait found an equivalent formulation of the FourColor Theorem in terms of edge 3coloring, stated here as Theorem 3. The next major contribution came in 1913 from G. D. Birkhoff, whose work allowed Franklin to prove in 1922 that the fourcolor conjecture is true for maps with at most twentyfive regions. The same method was used by other mathematicians to make progress on the fourcolor problem. Important here is the work by Heesch, who developed the two main ingredients needed for the ultimate proof—“reducibility ” and “discharging”. While the concept of reducibility was studied by other researchers as well, the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and he also conjectured that a suitable development of this method would solve the FourColor Problem. This was confirmed by Appel and Haken (abbreviated A&H) when they published their proof of the FourColor Theorem in two 1977 papers, the second one joint with Koch. An expanded version of the proof was later reprinted in
Linear Connectivity Forces Large Complete Bipartite Minors
, 2004
"... Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and ..."
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Cited by 28 (17 self)
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Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and minimum degree at least 31 2 (a + 1) − 3 are enough. The condition “a subdivision of Ka,sk ” is necessary since G could be a complete bipartite graph K31 where m could be arbitrarily 2 (a+1),m, large. The requirement on N(s, k, a) vertices is necessary since there exist graphs without Kaminor whose connectivity is Θ(a √ log a). When s = 1 and k = a, this implies that every 31 2 (a+1)connected graph with at least N(a) vertices has a Kaminor. This is the first result where a linear lower bound on the connectivity in terms of a forces a Kaminor. This was also conjectured in [68, 47, 69, 39]. Our result generalizes a recent result of Böhme and Kostochka [4] and resolves a conjecture of FonDerFlaass [16]. Our result together with a recent result in [25] also implies that there exists an absolute constant c such that there are only finitely many ckcontractioncritical graphs without Kk as a minor and there are only finitely many ckconnected ckcolorcritical graphs without Kkminors. These results are related to the wellknown conjecture of Hadwiger [17]. Our result was also motivated by the wellknown result of Erdős and Pósa [15]. Suppose that G is 31
The Colin de Verdière graph parameter
, 1997
"... In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless emb ..."
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In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.
A Relaxed Hadwiger's CONJECTURE FOR LIST COLORINGS
, 2005
"... Hadwiger’s Conjecture claims that any graph without Kk as a minor is (k − 1)colorable. It has been proved for k ≤ 6, and is still open for every k ≥ 7. It is not even known if there exists an absolute constant c such that any ckchromatic graph has Kk as a minor. Motivated by this problem, we show ..."
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Hadwiger’s Conjecture claims that any graph without Kk as a minor is (k − 1)colorable. It has been proved for k ≤ 6, and is still open for every k ≥ 7. It is not even known if there exists an absolute constant c such that any ckchromatic graph has Kk as a minor. Motivated by this problem, we show that there exists a computable constant f(k) such that any graph G without Kk as a minor admits a vertex partition V1,...,V⌈15.5k ⌉ such that each component in the subgraph induced on Vi (i ≥ 1) has at most f(k) vertices. This result is also extended to list colorings for which we allow monochromatic components of order at most f(k). When f(k) = 1, this is a coloring of G. Hence this is a relaxation of coloring and this is the first result in this direction.
Relaxing planarity for topological graphs
 Discrete and Computational Geometry, Lecture Notes in Comput. Sci., 2866
, 2003
"... Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be ..."
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Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straightline drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1
On the oddminor variant of Hadwiger’s conjecture
, 2011
"... A Klexpansion consists of l vertexdisjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be twocoloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains ..."
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A Klexpansion consists of l vertexdisjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be twocoloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd Klexpansion then its chromatic number is O(l √ log l). In doing so, we obtain a characterization of graphs which contain no odd Klexpansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertexdisjoint odd paths with endpoints in S, or there is a set X of at most 2k − 2 vertices such that every odd path with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results.
The Extremal Function for K9 Minors
, 2005
"... We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted. ..."
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We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted.
Counting planar graphs and related families of graphs
 In Surveys in combinatorics 2009, 169–210
, 2009
"... In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree dist ..."
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Cited by 8 (3 self)
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In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree distribution or the size of the largest kconnected component. Most of the results we present use generating functions and analytic tools.
Hadwiger's conjecture for quasiline graphs
, 2008
"... A graph G is a quasiline graph if for every vertex v ∈ V (G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasiline graphs is a proper superset of the class of line graphs. Hadwiger’s conjecture states that if a graph G is not tcolorable then it cont ..."
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Cited by 8 (4 self)
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A graph G is a quasiline graph if for every vertex v ∈ V (G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasiline graphs is a proper superset of the class of line graphs. Hadwiger’s conjecture states that if a graph G is not tcolorable then it contains Kt+1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour [10]. We extend their result to all quasiline graphs.