The Theorem of R.L. Brooks

Transcription

1 The Theorem of R.L. Brooks Talk by Bjarne Toft at the Yoshimi Egawa 60 Conference. Kagurazaka, Tokyo, Japan, September 10-14, 2013 Joint work with Michael Stiebitz.

2 Proc. Cambridge Phil. Soc. 1941

3 Colouring abstract graphs (rather than maps and graphs on surfaces) MILESTONES: A.B.Kempe ( ) 1879 K. Wagner ( ) 1937 R.L. Brooks ( ) 1941 H. Hadwiger ( ) 1943 G.A. Dirac ( ) 1951 T. Gallai ( ) 1963

4 Klaus Wagner

5 Hugo Hadwiger

6 Vierteljahrschr. der Naturf. Gesellschaft in Zürich 1943

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8 It does not seem quite right to me that the conjecture now is named after me. In my Zürich lecture I just took your starting point, that contained the decisive idea, and carried it over to general chromatic numbers.

9 Rowland Leonard Brooks ( ) Born February 6, 1916, in Lincolnshire, England Cambridge University 1935 Tax-inspector in London Brooks s famous note with his theorem was communicated to the Proc. Cambridge Phil. Soc. by W.T. Tutte in Nov and published in 1941 Died in London, June 18, 1993.

10 Further biographical information is not included 1993

11 Trinity Mathematical Society 1938

12 Trinity Four (Brooks, Smith, Stone, Tutte)

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14 Smith and Brooks met at Trinity in 1935 At the end of the first lecture Smith said to the young man sitting next to him: That was confusing. The young man answered: I thought it was a very good lecture. When is the next one? As for the next lecture the lecturer came in and started: IVANKEHIOSUTOKLSTMNDEJLSZIRTUNG The two realized after half an hour that they were perhaps in the wrong room. They were! They left together, and the young man introduced himself to Smith as Leonard Brooks. Leonard Brooks introduced Smith (and Stone) to a chess-playing friend (by the name Bill Tutte). A life-long friendship started!

15 Brooks Theorem 1941 Let G be a graph of maximum degree Δ, where Δ 2, and suppose that that no connected component of G is a complete (Δ + 1)-graph. Then G has chromatic number at most Δ. Let G be a graph of maximum degree 2, and suppose that that no connected component of G is an odd cycle. Then G has chromatic number 2. ALTERNATIVELY: Let G be a connected graph of maximum degree Δ. Then G has chromatic number Δ+1, and it has chromatic number equal to Δ+1 if and only if it is a complete (Δ+1)-graph or an odd cycle.

16 Wiley 2014?? Degree Bounds for the Chromatic Number Degeneracy and Colorings Orientations and Colourings Properties of Critical Graphs Critical Graphs with Few Edges Homomorphisms and Colorings Coloring hypergraphs Coloring Graphs on Surfaces Graph Coloring Problems

17 Gabriel Andrew Dirac CRITICAL GRAPHS WERE FIRST DEFINED IN G.A. DIRAC s PhD-thesis 1951

18 Dirac s Ph.D. Dissertation 1951

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20 Dirac proved Brooks Theorem independently

21 Dénes König Perhaps Graph Theory owes even more to the contact of human beings with human beings than to the contact of mankind with nature. GALLAI made the pictures for the book Vielleicht noch mehr als der Berührung der Menschheit mit der Natur verdankt die Graphentheorie der Berührung der Menschen untereinander.

22 König in Göttingen 1904/05

23 Proofs of Brooks Theorem Sequential colouring and colour-interchange (Brooks 1942) Sequential colouring (Lovász 1975) Kempe chains (Melnikov and Vizing 1969) Maximum independent sets and Δ-reduction (Gerencsér 1965; Catlin 1979; Tverberg 1983) Δ-Reduction (Rabern 2013) List-colourings (Erdős, Rubin and Taylor 1979) Critical graphs (Dirac 1951; Gallai 1963)

24 Sequential colouring

25 Δ-reduction (Landon Rabern - recent)

26 The case Δ = 3

27 List-colourings Let G be a connected graph, and let for each vertex v of G a list L(v) of at least d(v) different colours be given. Then G may be coloured such that each vertex gets a colour from its list, except if each block of G is either a complete graph or an odd cycle (G is a Gallai-tree). In fact: except if 1) L(v) = d(v) for all v 2) If G has only one block, then L(v) is the same for all v 3) G is a Gallai-tree

28 Proof of the list colour theorem Suppose that (G,L) is a bad pair NOT SATISFYING 1), 2) and 3). If u is a non-separating vertex of G and c L(u), then let L denote the lists obtained from L by removing c from L(u) and from L(v) for all neighbours v of u in G. Then (G-u, L ) is a bad pair SATISFYING 1), 2) and 3)

29 We assume that (G,L) is a bad pair We shall prove that G satisfes 1), 2) and 3) We know that G-u satisfies 1), 2) and 3)

30 Critical k-chromatic graphs BROOKS 1941: If G is k-critical (k 4) on n vertices (n>k) then 2e k 1 n + 1 DIRAC 1957: If G is k-critical (k 4) on n vertices (n>k) then 2e k 1 n + (k 3) with equality for n= 2k-1. GALLAI 1963: 2e k 1 n + ((k 3)/(k 2 3))n KOSTOCHKA & STIEBITZ 1999: If G is k-critical (k 4) on n vertices (n k+2 and n 2k-1) then 2e k 1 n + 2(k 3) with equality for n=2k

31 We all have a favorite paper - I have two (both exactly 50 years old):

32 Or rather: I have three favorites! Gallai s beautiful theory of alternating paths. The Gallai-Edmonds decomposition theorem. AN INTRIGUING FOOTNOTE : With the present methods I have succeeded in getting factorization theorems for general graphs besides σ=1 only for σ=2. I shall discuss these results on another occation

33 Smolenice June 1963

34 Tibor Gallai ( )

35 Critical graphs I The blocks in the minor subgraphs are complete and/or odd cycles (the minor graph is a forest of Gallai trees) This is best possible (by construction) If G is k-critical (k 4) on n vertices (n>k) then 2e k 1 n + ((k 3)/(k 2 3))n Gallai s Conjecture: 2e [(k 2 k 2)n k(k 3)]/(k 1)) and this is sharp for n=1 mod (k-1) Krivelevich 1997, Kostochka&Stiebitz 1999, 2000 and 2002, Kostochka&Yancey 2012

36 Critical graphs II A k-critical graph with 2k-2 vertices has disconnected complement The proof uses Gallai s theory of alternating paths from the 1950 paper Other proofs by Molloy 1999 and Stehlik 2003 The right minimum number of edges for all n at most 2k-1

37 Hajós Construction

38 Ore s Conjecture

39 f(k,n) = minimum number of edges in k-critical graph on n vertices (4 k n and n k+1) Dirac 1957: f(k,2k-1) known Gallai 1963: f(k, n) known for all n 2k-1 Ore 1967: f(k,n+k-1) f(k,n) + k(k-1)/2 1. Kostochka and Stiebitz 1999: f(k,2k) is known. Equality in Ore s inequality would therefore imply that f(k,n) is known for all values of k and n. Kostochka and Yancey 2013: f(k,n) is known for n = 1 mod (k-1) Fekete 1923: lim f(k,n)/n exists for all k, and it is known (2013). Kostochka and Yancey 2012: f(k,n) is known for k=4 and k=5.

40 f(4,n) = minimum number of edges in 4- critical graph on n vertices (4 n and n 5) f(4,n) = the integer part of 5n/3, i.e. f(4,n) = 5n/3 for n = 0 mod 3 f(4,n) = (5n-2)/3 for n = 1 mod 3 f(4,n) = (5n-1)/3 for n = 2 mod 3

41 Proof outline of f(4,n) = 5n/3

42 Grötzsch s Theorem (1959): Every planar triangle-free graph is 3-colourable.

43 4-critical graphs with empty minor graph

44 Planar 4-critical graphs without vertices of degree 3 (Koester 1984).

45 4-critical graphs with many edges/high min degree

46 4-critical graphs with all vertices of high degree ( max min δ(g) ) Simonovits and Toft 1971 Max min d(g) c 3 V G BEST POSSIBLE??

47 Critical k-chromatic graphs (on n vertices) with just one Major vertex Min max C c log n BEST POSSIBLE Alon, Krivelevich, Seymour 2000 Shapira&Thomas 2011 Max min C ~ c log n Erdős 1959 and 1962 Max min odd C??

48 Critical 4-chromatic graphs with long shortest odd cycles Max min odd C c n BEST POSSIBLE

49 Critical k-chromatic graphs with precisely two Major vertices IF there are precisely two major vertices and they are independent THEN the minor graph is disconnected Gallai s Conjecture: the number of conn. components in the minor graph is at least the number of components in the major graph PROVED by Stiebitz in 1982 USED by Krivelevich in 1997

50 A recent generalization by Landon Rabern Brooks Theorem: Every graph G with Δ(G) 3 satisfies (G) max{, Δ} Landon Rabern 2013: Every graph G with Δ(G) 3 satisfies (G) max{, Δ2, 5(Δ+1)/6}, where Δ2 is the maximum degree of a vertex v adjacent to another vertex of degree at least as large as the degree of v.

51 Unsolved Problem 1 Give the exact value of f(k,n) for all n k 4 and n k+1 I.e. prove or disprove Ore s Conjecture that f(k,n+k-1) = f(k,n) + k(k-1)/2 1

52 Unsolved Problem 2 The exact value of f(4,n) for all n 4 and n 5 is known. Describe all the extremal 4-critical graphs

53 Unsolved Problem 3 Determine the exact value of the minimum number of edges in a planar 4-critical graph on n vertices. Describe the extremal 4-critical graphs

54 All 4-critical graphs with at most 9 vertices

55 Unsolved Problem 4 Prove or disprove the existence of a constant c such that any 4- critical graph with minimum degree δ satisfies V(G) cδ 3.

56 Unsolved Problem 5 Borodin & Kostochka 1977, Catlin 1978, Lawrence 1978 proved that (G) Δ(G) (Δ+1)/( +1) provided 3 (G) Δ(G). It follows that (G) Δ(G) 2 provided 3 (G) (Δ(G) 2)/3. It follows that (G) Δ(G) 3 provided 3 (G) (Δ(G) 3)/4. OBTAIN BETTER/OPTIMAL CONDITIONS

57 Results by Reed (1999), Cranston and Rabern (2013), Farzad, Molloy and Reed (2005) = Δ = = Δ 13-3 = Δ-1 large -1 = Δ-2 large = -1 More general results by Molloy and Reed 2001 for large Δ

58 Unsolved Problem 6 Borodin & Kostochka s Conjecture 1977: (G) Δ(G) 1 provided (G) Δ(G) 1 and Δ(G) 9. I.e. : = Δ 9 = Proved for Δ(G) (Reed1999).

59 Unsolved Problem 7 Reed 1999 conjectured: (G) 1 2 (Δ(G) + 1) (G) (G) 2 3 (Δ(G) + 1) (G), provided Δ(G) 3

60 Unsolved Problem 8 Brooks Theorem: If k 3 and G is a graph, not containing the complete bipartite graph K(1, k+1), nor the complete (k+1)-graph Kk+1, then (G) k. Gyarfás Conjecture 1988: For any tree T there is a function f such that if G is a graph not containing T as an induced subgraph then (G) f( (G)).

61 Thank you for your attention. And congratulations, thanks and all best wishes to Yoshimi Egawa!