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Five-coloring plane graphs 229 For |V | = 3 this is obvious, since for the only uncolored vertex v we have |C(v)| ≥ 3, so there is a color available. Now we proceed by induction. Case 1: Suppose B has a chord, that is, an edge not in B that joins two vertices u, v ∈ B. The subgraph G 1 which is bounded by B 1 ∪ {uv} and contains x, y, u and v is near-triangulated and therefore has a 5-list coloring by induction. Suppose in this coloring the vertices u and v receive the colors γ and δ. Now we look at the bottom part G 2 bounded by B 2 and uv. Regarding u, v as pre-colored, we see that the induction hypotheses are also satisfied for G 2 . Hence G 2 can be 5-list colored with the available colors, and thus the same is true for G. B 1 u G 1 G 2 x y v B 2 Case 2: Suppose B has no chord. Let v 0 be the vertex on the other side of the α-colored vertex x on B, and let x, v 1 , . . . , v t , w be the neighbors of v 0 . Since G is near-triangulated we have the situation shown in the figure. Construct the near-triangulated graph G ′ = G\v 0 by deleting from G the vertex v 0 and all edges emanating from v 0 . This G ′ has as outer boundary B ′ = (B\v 0 ) ∪ {v 1 , . . . , v t }. Since |C(v 0 )| ≥ 3 by assumption (2) there exist two colors γ, δ in C(v 0 ) different from α. Now we replace every color set C(v i ) by C(v i )\{γ, δ}, keeping the original color sets for all other vertices in G ′ . Then G ′ clearly satisfies all assumptions and is thus 5-list colorable by induction. Choosing γ or δ for v 0 , different from the color of w, we can extend the list coloring of G ′ to all of G. □ v 0 w v t . .. v2 v 1 x(α) y(β) B So, the 5-list color theorem is proved, but the story is not quite over. A stronger conjecture claimed that the list-chromatic number of a plane graph G is at most 1 more than the ordinary chromatic number: Is χ l (G) ≤ χ(G) + 1 for every plane graph G ? Since χ(G) ≤ 4 by the four-color theorem, we have three cases: Case I: χ(G) = 2 =⇒ χ l (G) ≤ 3 Case II: χ(G) = 3 =⇒ χ l (G) ≤ 4 Case III: χ(G) = 4 =⇒ χ l (G) ≤ 5. α Thomassen’s result settles Case III, and Case I was proved by an ingenious (and much more sophisticated) argument by Alon and Tarsi. Furthermore, there are plane graphs G with χ(G) = 2 and χ l (G) = 3, for example the graph K 2,4 that we considered in the chapter on the Dinitz problem. But what about Case II? Here the conjecture fails: This was first shown by Margit Voigt for a graph that was earlier constructed by Shai Gutner. His graph on 130 vertices can be obtained as follows. First we look at the “double octahedron” (see the figure), which is clearly 3-colorable. Let α ∈ {5, 6, 7, 8} and β ∈ {9, 10, 11, 12}, and consider the lists that are given in the figure. You are invited to check that with these lists a coloring is not possible. Now take 16 copies of this graph, and identify all top vertices and all bottom vertices. This yields a graph on 16 · 8 + 2 = 130 vertices which {α, 1, 3, 4} {α, β,1, 2} {α, β,1, 2} {1, 2, 3, 4} {1, 2, 3, 4} β {α, 2, 3, 4} {β, 2, 3, 4} {β, 1, 3, 4}
230 Five-coloring plane graphs is still plane and 3-colorable. We assign {5, 6, 7, 8} to the top vertex and {9, 10, 11, 12} to the bottom vertex, with the inner lists corresponding to the 16 pairs (α, β), α ∈ {5, 6, 7, 8}, β ∈ {9, 10, 11, 12}. For every choice of α and β we thus obtain a subgraph as in the figure, and so a list coloring of the big graph is not possible. By modifying another one of Gutner’s examples, Voigt and Wirth came up with an even smaller plane graph with 75 vertices and χ = 3, χ l = 5, which in addition uses only the minimal number of 5 colors in the combined lists. The current record is 63 vertices. References [1] N. ALON & M. TARSI: Colorings and orientations of graphs, Combinatorica 12 (1992), 125-134. [2] P. ERDŐS, A. L. RUBIN & H. TAYLOR: Choosability in graphs, Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium 26 (1979), 125-157. [3] G. GONTHIER: Formal proof — the Four-Color Theorem, Notices of the AMS (11) 55 (2008), 1382-1393. [4] S. GUTNER: The complexity of planar graph choosability, Discrete Math. 159 (1996), 119-130. [5] N. ROBERTSON, D. P. SANDERS, P. SEYMOUR & R. THOMAS: The fourcolour theorem, J. Combinatorial Theory, Ser. B 70 (1997), 2-44. [6] C. THOMASSEN: Every planar graph is 5-choosable, J. Combinatorial Theory, Ser. B 62 (1994), 180-181. [7] M. VOIGT: List colorings of planar graphs, Discrete Math. 120 (1993), 215-219. [8] M. VOIGT & B. WIRTH: On 3-colorable non-4-choosable planar graphs, J. Graph Theory 24 (1997), 233-235.
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Martin Aigner Günter M. Ziegler Pr
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Prof. Dr. Martin Aigner FB Mathemat
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VI Preface to the Fourth Edition Wh
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VIII Table of Contents 21. A theore
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Six proofs of the infinity of prime
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Six proofs of the infinity of prime
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Bertrand’s postulate Chapter 2 We
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Bertrand’s postulate 9 times. Her
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Bertrand’s postulate 11 by compar
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Binomial coefficients are (almost)
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Binomial coefficients are (almost)
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Representing numbers as sums of two
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The law of quadratic reciprocity Ch
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The law of quadratic reciprocity 25
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Every finite division ring is a fie
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Every finite division ring is a fie
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Some irrational numbers Chapter 7
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Some irrational numbers 37 and this
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Some irrational numbers 39 F(x) may
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Some irrational numbers 41 Now assu
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44 Three times π 2 /6 v 1 1 2 y v
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46 Three times π 2 /6 For m = 1, 2
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48 Three times π 2 /6 1 f(t) = 1 t
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50 Three times π 2 /6 (3) It has b
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Hilbert’s third problem: decompos
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64 Lines in the plane, and decompos
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66 Lines in the plane, and decompos
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The slope problem Chapter 11 Try fo
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The slope problem 71 • Every move
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The slope problem 73 For the second
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76 Three applications of Euler’s
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82 Cauchy’s rigidity theorem Q: Q
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84 Cauchy’s rigidity theorem A be
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86 Touching simplices l l l ′ Pr
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88 Touching simplices For our examp
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90 Every large point set has an obt
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96 Borsuk’s conjecture Theorem. L
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98 Borsuk’s conjecture On the oth
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100 Borsuk’s conjecture To obtain
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Sets, functions, and the continuum
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In praise of inequalities Chapter 1
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In praise of inequalities 121 where
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In praise of inequalities 123 Proo
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In praise of inequalities 125 Seco
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128 The fundamental theorem of alge
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One square and an odd number of tri
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A theorem of Pólya on polynomials
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On a lemma of Littlewood and Offord
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On a lemma of Littlewood and Offord
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Cotangent and the Herglotz trick Ch
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Cotangent and the Herglotz trick 15
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Buffon’s needle problem Chapter 2
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Buffon’s needle problem 157 The c
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Combinatorics 25 Pigeon-hole and do
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162 Pigeon-hole and double counting
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Tiling rectangles Chapter 26 Some m
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Tiling rectangles 175 Third proof.
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Tiling rectangles 177 References [1
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