The Rainbow Turán Problem for Even Cycles

Transcription

1 The Rainbow Turán Problem for Even Cycles Shagnik Das University of California, Los Angeles Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov

2 Plan 1 Historical Background Turán Problems Colouring Problems 2 Rainbow Turán Problem Definition Motivation Known Results 3 Our Results Summary Warm Up Sketch of Proof

3 Turán s Theorem Mantel s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n2 4 edges.

4 Turán s Theorem Mantel s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n2 4 edges. Turán generalised to cliques of any order Theorem (Turán, 1941) If a graph ( G on n vertices ) has no clique of order r, then G has at most 1 1 r 1 + o(1) n 2 2 edges.

5 Turán s Theorem: Extended Can define Turán numbers of general graphs Definition (Turán numbers) Given any graph H, we define the Turán number ex(n, H) to be the maximum number of edges in an H-free graph on n vertices.

6 Turán s Theorem: Extended Can define Turán numbers of general graphs Definition (Turán numbers) Given any graph H, we define the Turán number ex(n, H) to be the maximum number of edges in an H-free graph on n vertices. Erdős and Stone found asymptotics for all non-bipartite graphs Theorem (Erdős-Stone, 1946) For all graphs H, lim n ex(n, H)/ ( n 2) = 1 (χ(h) 1) 1.

7 Turán s Theorem: Open Problems Turán problem for bipartite graphs generally open

8 Turán s Theorem: Open Problems Turán problem for bipartite graphs generally open Particularly interesting is the case of even cycles

9 Turán s Theorem: Open Problems Turán problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) ( ) For all k 2, ex(n, C 2k ) = O n 1+ 1 k.

10 Turán s Theorem: Open Problems Turán problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) ( ) For all k 2, ex(n, C 2k ) = O n 1+ 1 k. Matching lower bound only known for k = 2, 3, 5

11 Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l 1, there exists R(k, l) such that any red-blue colouring of K R(k,l) contains either a red K k or a blue K l.

12 Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l 1, there exists R(k, l) such that any red-blue colouring of K R(k,l) contains either a red K k or a blue K l. Determining the Ramsey numbers R(k, l) a widely open problem

13 Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l 1, there exists R(k, l) such that any red-blue colouring of K R(k,l) contains either a red K k or a blue K l. Determining the Ramsey numbers R(k, l) a widely open problem Introduction of the probabilistic method

14 Colouring Problems: Extensions Theorem (Erdős-Rado, 1950) For every t, there is an n such that every edge-colouring of K n has a copy of K t with one of the following canonical colourings: constant rainbow minimum maximum

15 Colouring Problems: Extensions Theorem (Erdős-Rado, 1950) For every t, there is an n such that every edge-colouring of K n has a copy of K t with one of the following canonical colourings: constant rainbow minimum maximum In particular, for every t there is an n such that every proper edge-colouring of K n has a rainbow K t.

16 Rainbow Turán Problem First introduced by Keevash, Mubayi, Sudakov, Verstraëte

17 Rainbow Turán Problem First introduced by Keevash, Mubayi, Sudakov, Verstraëte Definition (Rainbow Turán Numbers) Given a graph H, we define the rainbow Turán number ex (n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H.

18 Rainbow Turán Problem First introduced by Keevash, Mubayi, Sudakov, Verstraëte Definition (Rainbow Turán Numbers) Given a graph H, we define the rainbow Turán number ex (n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H. Trivial bound: ex(n, H) ex (n, H)

19 Rainbow Turán Problem First introduced by Keevash, Mubayi, Sudakov, Verstraëte Definition (Rainbow Turán Numbers) Given a graph H, we define the rainbow Turán number ex (n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H. Trivial bound: ex(n, H) ex (n, H) Reduction to regular Turán problem ex (n, H) ex(n, H) + o(n 2 )

20 Bk -sets: Motivation Definition (B k -sets) A subset A of an abelian group G is a B k -set if every g G has at most one representation of the form g = a 1 + a a k, a i A.

21 Bk -sets: Motivation Definition (B k -sets) A subset A of an abelian group G is a B k -set if every g G has at most one representation of the form g = a 1 + a a k, a i A. Definition (B k -sets) A subset A of an abelian group G is a B k -set if there are no two disjoint k-sets {x 1, x 2,..., x k }, {y 1, y 2,..., y k } A with the same sum.

22 Bk -sets: Motivation Definition (B k -sets) A subset A of an abelian group G is a B k -set if every g G has at most one representation of the form g = a 1 + a a k, a i A. Definition (B k -sets) A subset A of an abelian group G is a Bk -set if there are no two disjoint k-sets {x 1, x 2,..., x k }, {y 1, y 2,..., y k } A with the same sum. Example (B k -sets need not be B k-sets) Let k = 3, G = Z/6Z, and A = {0, 1, 2, 3, 4, 5}. A is not a B 3 -set: = = A is a B3 -set: the sum of all six elements is odd.

23 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs

24 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs Construct bipartite graph on X Y, where X, Y = G Edge (x, y) iff y x = a A G, colour a

25 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs Construct bipartite graph on X Y, where X, Y = G Edge (x, y) iff y x = a A G, colour a x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 Example with G = Z/5Z, A = {1, 3}

26 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs Construct bipartite graph on X Y, where X, Y = G Edge (x, y) iff y x = a A G, colour a x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 Example with G = Z/5Z, A = {1, 3}

27 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs Construct bipartite graph on X Y, where X, Y = G Edge (x, y) iff y x = a A G, colour a x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 Example with G = Z/5Z, A = {1, 3}

28 Bk -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B k -sets rainbow-c 2k-free bipartite graphs Construct bipartite graph on X Y, where X, Y = G Edge (x, y) iff y x = a A G, colour a x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 Example with G = Z/5Z, A = {1, 3} Rainbow C 2k x 1 y 1 x 2 y 2... x k y k B = {y 1 x 1,..., y k x k } and C = {y 1 x 2, y 2 x 3,..., y k x 1 }

29 Bk -sets: Known Results A G has ( A +k 1) k different k-sums ( ) if G = n, A a B k -set, then A = O n 1 k

30 Bk -sets: Known Results A G has ( A +k 1) k different k-sums ( ) if G = n, A a B k -set, then A = O n 1 k ( ) Bose and Chowla: B k -sets of size Ω n 1 k in G = Z/nZ ) Z/nZ has Bk (n -sets of size Ω 1 k

31 Bk -sets: Known Results A G has ( A +k 1) k different k-sums ( ) if G = n, A a B k -set, then A = O n 1 k ( ) Bose and Chowla: B k -sets of size Ω n 1 k in G = Z/nZ ) Z/nZ has Bk (n -sets of size Ω 1 k ( ) ex (n, C 2k ) = Ω n 1+ 1 k

32 Bk -sets: Known Results A G has ( A +k 1) k different k-sums ( ) if G = n, A a B k -set, then A = O n 1 k ( ) Bose and Chowla: B k -sets of size Ω n 1 k in G = Z/nZ ) Z/nZ has Bk (n -sets of size Ω 1 k ( ) ex (n, C 2k ) = Ω n 1+ 1 k ( ) Ruzsa: O n 1 k analytic upper bound for Bk -sets Bound on ex (n, C 2k ) would give a combinatorial proof

33 Rainbow Turán Problem: Bipartite Graphs If H has maximum ( degree ) s in one part, then ex (n, H) = O n 2 1 s

34 Rainbow Turán Problem: Bipartite Graphs If H has maximum ( degree ) s in one part, then ex (n, H) = O n 2 1 s Gives correct order ( of ) magnitude for 4-cycle: ex (n, C 4 ) = O n

35 Rainbow Turán Problem: Bipartite Graphs If H has maximum ( degree ) s in one part, then ex (n, H) = O n 2 1 s Gives correct order ( of ) magnitude for 4-cycle: ex (n, C 4 ) = O n Analysing rainbow paths ex (n, C 6 ) = O ( ) n

36 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be?

37 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be? Hypercube construction Ω(n log n)

38 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be? Hypercube construction Ω(n log n) ( ) 6-cycle bound O n

39 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be? Hypercube construction Ω(n log n) ( ) 6-cycle bound O n Two interesting questions:

40 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be? Hypercube construction Ω(n log n) ( ) 6-cycle bound O n Two interesting questions: 1. Determine size of rainbow acyclic graphs

41 Rainbow Turán Problem: Rainbow Acyclic Graphs How large can an n-vertex graph without rainbow cycles be? Hypercube construction Ω(n log n) ( ) 6-cycle bound O n Two interesting questions: 1. Determine size of rainbow acyclic graphs 2. Determine asymptotics of ex (n, C 2k )

42 Our Results

43 Our Results Theorem (D.-Lee-Sudakov, 2012+) A rainbow ( acyclic n-vertex graph can have at most n exp (log n) 1 +η) 2 edges for any η > 0.

44 Our Results Theorem (D.-Lee-Sudakov, 2012+) A rainbow ( acyclic n-vertex graph can have at most n exp (log n) 1 +η) 2 edges for any η > 0. Theorem (D.-Lee-Sudakov, 2012+) For every integer k 2, ex (n, C 2k ) = O where ε k 0 as k. ) (n 1+ (1+ε k ) ln k k,

45 Our Proof: Warm Up The black-and-white version of our theorem

46 Our Proof: Warm Up The black-and-white version of our theorem Proposition If G on n vertices has more than n 1+ε edges, then it has a cycle of length at most 2 ε.

47 Our Proof: Warm Up The black-and-white version of our theorem Proposition If G on n vertices has more than n 1+ε edges, then it has a cycle of length at most 2 ε. Proof by induction on n Base case: For n 2 ε, result follows from existence of cycle

48 Our Proof: Warm Up The black-and-white version of our theorem Proposition If G on n vertices has more than n 1+ε edges, then it has a cycle of length at most 2 ε. Proof by induction on n Base case: For n 2 ε, result follows from existence of cycle Induction step: May assume minimum degree δ > n ε

49 Our Proof: Warm Up II v 0 Choose an arbitrary vertex v 0 G

50 Our Proof: Warm Up II N 1 v 0 > n ε Expand its neighbourhood N 1

51 Our Proof: Warm Up II N 1 N 2 v 0 > n ε > n 2ε Expand the second neighbourhood N 2

52 Our Proof: Warm Up II N 1 N 2 N 1 ε v 0... > n ε > n 2ε > n If no short cycle, then we will eventually exceed n vertices - contradiction

53 Our Proof: A Sketchy Overview Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n 1+ε edges has a rainbow cycle of length at most L(ε).

54 Our Proof: A Sketchy Overview Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n 1+ε edges has a rainbow cycle of length at most L(ε). Proof idea: As before, may assume minimum degree δ(g) > C(ε)n ε

55 Our Proof: A Sketchy Overview Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n 1+ε edges has a rainbow cycle of length at most L(ε). Proof idea: As before, may assume minimum degree δ(g) > C(ε)n ε Will grow a subtree T out of an arbitrary vertex v 0

56 Our Proof: A Sketchy Overview II v 0 Choose an arbitrary vertex v 0 G

57 Our Proof: A Sketchy Overview II L 1 v 0 n α 1 Induction implies the small levels must expand

58 Our Proof: A Sketchy Overview II L 1 L 2 v 0 n α 1 n α 2 Induction implies the small levels must expand

59 Our Proof: A Sketchy Overview II L 1 L 2 L 3 v 0 n α 1 n α 2 n α 3 Induction implies the small levels must expand

60 Our Proof: A Sketchy Overview II L 1 L 2 L 3 v 0 n α 1 n α 2 n α 3 We ensure all vertices have a rainbow path back to v 0

61 Our Proof: A Sketchy Overview II L 1 L 2 L 3 v 0 n α 1 n α 2 n α 3 Let X i,c L i be those with colour c in their paths.

62 Our Proof: A Sketchy Overview II L i n α i Growing next level: consider unused neighbours of L i

63 Our Proof: A Sketchy Overview II L i L i+1 n α i n α i+1 Restrict L i+1 to those with between d and 2d neighbours in L i

64 Our Proof: A Sketchy Overview II L i L i+1 n ε d n α i n α i+1 Must have d n ε+α i α i+1

65 Our Proof: A Sketchy Overview II L i L i+1 n α i n α i+1 If no short rainbow cycle, then neighbours in L i must share a colour

66 Our Proof: A Sketchy Overview II L i L i+1 n α i n α i+1 If no short rainbow cycle, then neighbours in L i must share a colour

67 Our Proof: A Sketchy Overview II L i L i+1 n α i n α i+1 Let W c be those in L i+1 with many neighbours in X i,c

68 Our Proof: A Sketchy Overview II Restrict to G[X i,c, W c ]

69 Our Proof: A Sketchy Overview II n ε d If d n ε, short rainbow cycle by induction

70 Our Proof: A Sketchy Overview II n ε d Thus d n ε n α i+1 n α i - desired expansion

71 Open Problems Question (Rainbow Acyclicity) We know the largest properly edge-coloured ( n-vertex graph has at least Ω(n log n) edges, and at most n exp (log n) 1 +o(1)) 2 edges. Can the gap be narrowed? Question (Rainbow Turán Number for Even Cycles) ( ) We have the bounds Ω n 1+ 1 k ex (n, C 2k ) O (n Can this gap be narrowed? 1+ (1+o(1)) ln k k ).

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