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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
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Progress on Perfect Graphs
, 2003
"... A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restr ..."
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Cited by 9 (3 self)
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A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first
EXCLUDING MINORS IN NONPLANAR GRAPHS OF GIRTH AT LEAST FIVE
, 1999
"... A graph is quasi 4connected if it is simple, 3connected, has at least five vertices, and for every partition (A, B, C) of V(G) either C≥4, or G has an edge with one end in A and the other end in B, orone of A,B has at most one vertex. We show that any quasi 4connected nonplanar graph with minim ..."
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A graph is quasi 4connected if it is simple, 3connected, has at least five vertices, and for every partition (A, B, C) of V(G) either C≥4, or G has an edge with one end in A and the other end in B, orone of A,B has at most one vertex. We show that any quasi 4connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P − 10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte’s Four Flow Conjecture: every 2edge connected graph with no minor isomorphic to P − 10 has a nowherezero 4flow. This extends a result of Kilakos and Shepherd who proved the same for 3regular graphs.
Recent Excluded Minor Theorems
 SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.