A dual to the Turán problem

Transcription

1 Middlebury College joint work with Ron Gould (Emory University) Tomasz Luczak (Adam Mickiewicz University and Emory University) Oleg Pikhurko (Carnegie Mellon University) October 008 Dartmouth College Combinatorics Seminar

2 Definition A graph G is F-saturated if F G and F G + e for any e E(G).

3 The father of Extremal Graph Theory Paul Turán Turán s Theorem, 1941 The maximum number of edges in a K t -saturated graph on n vertices, ex(n,k t ), is (1 1 t 1 )n. Furthermore, the only graph which achieves this bound is K n1,n,...n t 1, where the n i are as balanced as possible.

4 Erdős-Stone-Simonovits Theorem Theorem Given a graph F with chromatic number, χ(f), at least three, the maximum number of edges in an F-saturated graph on n vertices, ex(n,f), is (1 1 χ(f) 1 )n + o(n ).

5 Problem Determine the minimum number of edges of an F-saturated graph, denoted by sat(n, F).

6 Theorem (Erdős, Hajnal, Moon ) ( ) t sat(n,k t ) = (t )(n 1) Furthermore, the only K t -saturated graph with this many edges is K t + K n t+.

7 Theorem (Ollmann - 7, Tuza - 86) sat(n,c 4 ) = 3n 5, n 5

8 Theorem (Ollmann - 7, Tuza - 86) sat(n,c 4 ) = 3n 5, n 5

9 Theorem (Ollmann - 7, Tuza - 86) sat(n,c 4 ) = 3n 5, n 5.

10 Theorem (Fisher, Fraughnaugh, Langley - 97) Theorem sat(n,p 3 connected) = 3n 5. sat(n,c 5 ) = 10n 10,n 1. 7 Fisher, Fraughnaugh, Langley, - 95 gave the upper bound. Y.C.Chen gave the lower bound. Problem (FFL) Determine sat(n,p 4 connected).

11 Hamiltonian Cycles Theorem sat(n,c n ) = 3n + 1,n 53. Bondy ( 7) showed the lower bound. Clark, Entringer, Crane and Shapiro ( 83-86) gave upper bound based on Isaacs flower snarks (girth 5, 6). L. Stacho ( 96) gave further constructions based on the Coxeter graph (girth 7).

12 Hamiltonian Cycles Theorem sat(n,c n ) = 3n + 1,n 53. Bondy ( 7) showed the lower bound. Clark, Entringer, Crane and Shapiro ( 83-86) gave upper bound based on Isaacs flower snarks (girth 5, 6). L. Stacho ( 96) gave further constructions based on the Coxeter graph (girth 7). Problem (Horák, Širáň - 86) Is there a maximally non-hamiltonian graph of girth at least 8?

13 Conjecture (Bollobás - 78) There exist constants, c 1,c, such that n + c 1 n l sat(n,c l ) n + c n l. Theorem (Barefoot, Clark, Entringer, Porter, Székely, Tuza - 96) (1 + 1 l + 8 )n sat(n,c l)

14 Theorem (Barefoot et al. - 96) sat(n,c l ) (1 + 6 l 3 )n + O(l ) for l odd, l 9 sat(n,c l ) (1 + 4 l )n + O(l3 ) for l even, l 14

15 Theorem (Barefoot et al. - 96) [Gould, Luczak, S. - 06] sat(n,c l ) ( l 3 l 3 )n + 4 for l odd, l 9, l 17,n 7l sat(n,c l ) ( l )n + 5l 4 for l even l 14, l 10,n 3l Theorem [Gould, Luczak, S. - 06] For l = 8, 9, 11, 13 or 15 and n l sat(n,c l ) < 3n + l

16 Our Inspiration Lance Armstrong s Trek Madone SSL proto and the complete special edition Bontrager front wheel with super-minimal 19mm tubulars.

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20 The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ

21 The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ

22 The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ

23 The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ

24 Counting Edges of the Luczak Wheel For l = k + and n a mod k, E(L Wheel) = Rim {}}{ (n k a)+ Spokes {}}{ n k a k Flange {}}{ + a + Hub ({}}{ ) k. Theorem [GLS] For k 4, l = k +, n a mod k and n 3l, sat(n,c l ) n(1 + 1 k ) + k 3k n(1 + l ) + 5l 4. a k

25 The Odd Luczak Wheel, l = k Ê Ñ Ò «Ö ½ Ö ¾ «Ð Ò ÀÙ «Ö ½ Ò ½ ËÔÓ Ã Ã ËÔÓ ¹ÒÙØ

26 C l -saturated graphs of minimum size l sat(n,c l ) n Reference 3 = n 1 3 EHM 4 = 3n 5 5 Ollmann; Tuza 5 = 10n FFL; Chen 6 3n 11 Barefoot et al. 7 7n Barefoot et al. 8,9,11,13,15 3n + l l GLS 10 and 0 mod ( 1 + ) l n + 5l 4 3l Luczak wheel 17 and 1 mod ( 1 + ) l 3 n + 5l 4 7l Luczak wheel n 3n+1 0 Bondy; CE, CES

27 Problem (Barefoot et al. - 96) Determine the value of l which minimizes sat(n,c l ) for fixed n. Problem Are any of these constructions optimal? Can one improve the lower bound?

28 Other Subgraphs Other values of sat(n,f) known for: matchings (Mader - 73), stars and paths (Kászonyi and Tuza - 86), hamiltonian path, P n (Frick and Singleton, 05; Dudek, Katona, Wojda - 06) longest path = detour(beineke, Dunbar, Frick, 05) disjoint cliques of the same order(faudree, Gould, Jacobson, Ferrara - 08)

29 Difficulties and Hereditary Properties Lacking Quote from Erdős, Hajnal and Moon: One of the difficulties of proving these conjectures may be that the obvious extremal graphs are certainly not unique, which fact may make an induction proof difficult. sat(n,f) sat(n + 1,F) F 1 F sat(n, F 1 ) sat(n, F ) F F sat(n,f ) sat(n,f)

30 Difficulties and Hereditary Properties Lacking Quote from Erdős, Hajnal and Moon: One of the difficulties of proving these conjectures may be that the obvious extremal graphs are certainly not unique, which fact may make an induction proof difficult. sat(n,f) sat(n + 1,F) F 1 F sat(n, F 1 ) sat(n, F ) F F sat(n,f ) sat(n,f) sat(k 1,P 4 ) = k + 1 and sat(k,p 4 ) = k sat(n, {P 5,S 4 }) = n 1 > sat(n,p 5 ) sat(n,k 4 ) = n 3 but sat(n,k 5 S 3 ) 3 n

31 Best known upper bound Theorem (Kászonyi and Tuza - 86 ) Let F be a graph. Set Then u = V (F) α(f) 1 s = min{e(f ) : F F,α(F ) = α(f), V (F ) = α(f) + 1}. sat(n,f) (u + s 1 u(s + u) )n. They considered a clique on u vertices joined to an (s 1)-regular graph.

32 Best Known Lower Bound

33 Best Known Lower Bound????

34 Best Known Lower Bound Problem For an arbitrary graph F, determine a non-trivial lower bound on sat(n,f).

35 Let sat(n,f,δ) equal minimum number of edges in a graph on n vertices and minimum degree δ that is F-saturated. Theorem (Duffus, Hanson - 86) sat(n,k 3,) = n 5, n 5. sat(n,k 3,3) = 3n 15, n 10. Problem (Bollobás - 95) Is it true that for every fixed δ 1 one has sat(n,k 3,δ) = δn O(1)?

36 Theorem (Gould, S. - 07) For integers t 3, n 4t 4, sat(n,k t() ) sat(n,k t(),t 3) = (4t 5)n 4t + 6t 1. Problem Given a fixed graph F, for n sufficiently large determine if the function sat(n,f,δ) is monotonically increasing in δ.

37 Theorem (Pikhurko, S. - 08) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.

38 Theorem (Pikhurko, S. - 08) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.

39 Theorem (Pikhurko, S. - 08) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.

40 Proof of Lower Bound Let G be a K,3 -saturated graph.

41 Proof of Lower Bound Let G be a K,3 -saturated graph. If δ(g) 4, then E(G) n and we are done.

42 Proof of Lower Bound Let G be a K,3 -saturated graph. If δ(g) = 1 then,

43 Proof of Lower Bound If δ(g) = 1 then,

44 Proof of Lower Bound If δ(g) = 1 then, and so E(G) n 3.

45 Proof of Lower Bound Otherwise, δ(g) 3, pick vertex of minimum degree and consider breadth-first search tree. V1 V V3

46 Proof of Lower Bound Otherwise, δ(g) 3, pick vertex of minimum degree and consider breadth-first search tree. V1 V V3 Tree has n 1 edges, we must find n Cn 3/4 more edges.

47 Divide and Conquer V1 V V3 Y0 Y Y 1

48 Divide and Conquer V1 V V3 Y0 Y Y 1

49 Hit em where they re weakest V1 V V3 Y0 Y Y 1 Y 0 has at most one component which is a tree. Pick up an extra V 3 1 edges.

50 More Division and More Conquering V1 V X 0 X 1 X V3 Pick up extra V #(trees in X 0 ) edges.

51 More Division and More Conquering V1 V X 0 X X 1 V3

52 More Hitting Weak Spots V1 V X 0 X X 1 V3 Trees in X 0 are connected via a path of length at most three through V 3.

53 More Hitting Weak Spots V1 V X 0 X 1 X small degree vertices large degree vertices V3 Small degree vertices can only serve so many trees of X 0. So, sum of large degree vertices is large.

54 More Hitting Weak Spots V1 V X 0 X 1 X small degree vertices large degree vertices V3 Small degree vertices can only serve so many trees of X 0. So, sum of large degree vertices is large. This allows us to add #(trees in X 0 ) O(n 3/4 ) edges to the count. Completes proof.

55 Recently, Bohman, Fonoberova and Pikhurko have announced the determination of sat(n,k s1,s,...,s r ). sat(n,k s1,s,...,s r ) = (s 1 + s s r s r 1 where s 1 s... s r. + o(1))n,

56 And Ramsey Numbers F (F 1,...,F t ) if any t coloring of E(F) contains a monochromatic F i -subgraph of color i for some i [t]. Conjecture (Hanson and Toft, 87) Given t and numbers m i 3,i [t], let F = {F : F (K m1,...,k mt )}. Let r = r(k m1,...,k mt ) be the classical Ramsey number. Then ( ) r sat(n, F) = (r )(n 1).

57 Many Thanks!! Talk and results are available online at: jschmitt/