On subgraphs of large girth

Transcription

1 1/34 On subgraphs of large girth In Honor of the 50th Birthday of Thomas Vojtěch Rödl joint work with Domingos Dellamonica May, 2012

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7 7/34 ARRANGEABILITY AND CLIQUE SUBDIVISIONS Vojtěch Rödl* Department of ematics and Computer Science Emory University Atlanta, GA and Thomas** School of ematics Georgia Institute of Technology Atlanta, GA ABSTRACT Let k be an integer. A graph G is k-arrangeable (concept introduced by Chen and Schelp) if the vertices of G can be numbered v1, v2,..., vn in such a way that for every integer i with 1 i n, at most k vertices among {v1, v2,..., vi} have a neighbor v {vi+1, vi+2,..., vn} that is adjacent to vi. We prove that for every integer p 1, if a graph G is not p 8 - arrangeable, then it contains a Kp-subdivision. By a result of Chen and Schelp this implies that graphs with no Kp-subdivision have linearly bounded Ramsey numbers, and by a result of Kierstead and Trotter it implies that such graphs have bounded game chromatic number.

8 8/34 An old question of Erdős and Hajnal Is it true that for every k and g there exists χ = χ(k, g) such that any graph G with χ(g) χ contains a subgraph H G with χ(h) k and girth(h) g?

9 8/34 An old question of Erdős and Hajnal Is it true that for every k and g there exists χ = χ(k, g) such that any graph G with χ(g) χ contains a subgraph H G with χ(h) k and girth(h) g? R (1977): True for g = 4, i.e., triangle-free subgraph H with large χ(h).

10 9/34 Thomassen s conjecture Let d(g) = 2e(G) v(g) denote the average degree of G. Conjecture (1983) For every k and g there exists D = D(k, g) such that any graph G with d(g) D contains a subgraph H G with d(h) k and girth(h) g.

11 9/34 Thomassen s conjecture Let d(g) = 2e(G) v(g) denote the average degree of G. Conjecture (1983) For every k and g there exists D = D(k, g) such that any graph G with d(g) D contains a subgraph H G with d(h) k and girth(h) g. Note: trivial to get rid of odd cycles by taking a bipartite subgraph H.

12 10/34 Known results Pyber, R, and Szemerédi (1995): true for all graphs G satisfying (G) e α k,g d(g).

13 10/34 Known results Pyber, R, and Szemerédi (1995): true for all graphs G satisfying (G) e α k,g d(g). Kühn and Osthus (2004): true for g 6, i.e., can obtain bipartite subgraph H which is C 4 -free;

14 10/34 Known results Pyber, R, and Szemerédi (1995): true for all graphs G satisfying (G) e α k,g d(g). Kühn and Osthus (2004): true for g 6, i.e., can obtain bipartite subgraph H which is C 4 -free; Dellamonica, Koubek, Martin, R. (2011): an alternative proof of the above result (next slides).

15 10/34 Known results Pyber, R, and Szemerédi (1995): true for all graphs G satisfying (G) e α k,g d(g). Kühn and Osthus (2004): true for g 6, i.e., can obtain bipartite subgraph H which is C 4 -free; Dellamonica, Koubek, Martin, R. (2011): an alternative proof of the above result (next slides). Dellamonica, R. (2011): true for all C 4 -free graphs G satisfying (G) e eβ k,g d(g) (this talk).

16 10/34 Known results Pyber, R, and Szemerédi (1995): true for all graphs G satisfying (G) e α k,g d(g). Kühn and Osthus (2004): true for g 6, i.e., can obtain bipartite subgraph H which is C 4 -free; Dellamonica, Koubek, Martin, R. (2011): an alternative proof of the above result (next slides). Dellamonica, R. (2011): true for all C 4 -free graphs G satisfying (G) e eβ k,g d(g) (this talk).

17 11/34 Alternative proof for g 6 Definition (Intersection Pattern) For k-partite k-graph H on V = V 1 V k, the intersection pattern of distinct e, f H is {i [k] : e V i = f V i }. Let P(H) denote the set of intersection patterns of all e f H. J = {2, 3} is an intersection pattern {

18 11/34 Alternative proof for g 6 Definition (Intersection Pattern) For k-partite k-graph H on V = V 1 V k, the intersection pattern of distinct e, f H is {i [k] : e V i = f V i }. Let P(H) denote the set of intersection patterns of all e f H. Definition (Star, Kernel) A hypergraph S is a star with kernel K if for all distinct e, f S, e f = K.

19 12/34 Alternative proof for g 6 Definition (Strong) A k-partite k-graph H is called t-strong if for every J P(H ) and e H there is a star S H, containing e, with kernel e J and S = t.

20 12/34 Alternative proof for g 6 Definition (Strong) A k-partite k-graph H is called t-strong if for every J P(H ) and e H there is a star S H, containing e, with kernel e J and S = t. J = {2, 3} is an intersection pattern {

21 12/34 Alternative proof for g 6 Definition (Strong) A k-partite k-graph H is called t-strong if for every J P(H ) and e H there is a star S H, containing e, with kernel e J and S = t. e J = {v 2, v 3 } {

22 12/34 Alternative proof for g 6 Definition (Strong) A k-partite k-graph H is called t-strong if for every J P(H ) and e H there is a star S H, containing e, with kernel e J and S = t. A t-star for t = 3 {

23 13/34 Alternative proof for g 6 Lemma (Füredi, 1983) For every k, t, there exists c = c(k, t) such that any k-graph H contains a t-strong subhypergraph H with H c H.

24 14/34 A common trick: take a bipartite subgraph From arbitrary G 0 with d(g 0 ) D k, get G G 0 with δ = δ(g) d(g 0 )/2.

25 14/34 A common trick: take a bipartite subgraph From arbitrary G 0 with d(g 0 ) D k, get G G 0 with δ = δ(g) d(g 0 )/2.

26 14/34 A common trick: take a bipartite subgraph From arbitrary G 0 with d(g 0 ) D k, get G G 0 with δ = δ(g) d(g 0 )/2.

27 14/34 A common trick: take a bipartite subgraph From arbitrary G 0 with d(g 0 ) D k, get G G 0 with δ = δ(g) d(g 0 )/2.

28 15/34 A common trick: take a bipartite subgraph Repeat until no more edge between vertices of degree > δ.

29 15/34 A common trick: take a bipartite subgraph First case: at least half of the edges are red = G red is almost regular, i.e. (G red ) δ = O(d(G red )).

30 15/34 A common trick: take a bipartite subgraph First case: at least half of the edges are red = G red is almost regular, i.e. (G red ) δ = O(d(G red )). Pyber-R-Szemerédi: G red contains subgraph H with d(h) k and girth(h) g.

31 15/34 A common trick: take a bipartite subgraph Second case: at least half of the edges are purple (crossing). Then set A = red set, B = blue set and G = (A, B; E) = G purple.

32 16/34 Applying Füredi s Lemma A little more work yields a bipartite graph G = (A, B; E) where deg G (v) = d k for all v A and, N G (v) N G (w) for all v w A.

33 16/34 Applying Füredi s Lemma A little more work yields a bipartite graph G = (A, B; E) where deg G (v) = d k for all v A and, N G (v) N G (w) for all v w A. Define H as a d-graph with V (H) = B and edge set { NG (v) : v A }.

34 16/34 Applying Füredi s Lemma A little more work yields a bipartite graph G = (A, B; E) where deg G (v) = d k for all v A and, N G (v) N G (w) for all v w A. Define H as a d-graph with V (H) = B and edge set { NG (v) : v A }. Apply Füredi s Lemma with some large t: H H is d-partite and t-strong. Note H A A.

35 17/34 The hypergraph from Füredi s Lemma

36 18/34 Dichotomy Recall P(H ) 2 [d] is the family of all intersection patterns of H. Let Γ be the shadow graph of P(H ): ( Γ = [d], { {i, j} : J P(H ), J {i, j} }).

37 18/34 Dichotomy Recall P(H ) 2 [d] is the family of all intersection patterns of H. Let Γ be the shadow graph of P(H ): ( Γ = [d], { {i, j} : J P(H ), J {i, j} }). Either 1 Γ has a large independent set I, or

38 18/34 Dichotomy Recall P(H ) 2 [d] is the family of all intersection patterns of H. Let Γ be the shadow graph of P(H ): ( Γ = [d], { {i, j} : J P(H ), J {i, j} }). Either 1 Γ has a large independent set I, or 2 (Γ), the max degree of Γ, is large.

39 19/34 Alternative Proof for g = 6 1 Γ has a large independent set I = the desired subgraph H (d(h) I k and girth(h) 6).

40 19/34 Alternative Proof for g = 6 1 Γ has a large independent set I = the desired subgraph H (d(h) I k and girth(h) 6). Claim. If B = B 1 B d are the classes of H, then H = G [ A, i I B i] is C4 -free.

41 19/34 Alternative Proof for g = 6 1 Γ has a large independent set I = the desired subgraph H (d(h) I k and girth(h) 6). Claim. If B = B 1 B d are the classes of H, then H = G [ A, i I B i] is C4 -free. Proof. If not...

42 19/34 Alternative Proof for g = 6 1 Γ has a large independent set I = the desired subgraph H (d(h) I k and girth(h) 6). Claim. If B = B 1 B d are the classes of H, then H = G [ A, i I B i] is C4 -free. Proof. If not...

43 19/34 Alternative Proof for g = 6 1 Γ has a large independent set I = the desired subgraph H (d(h) I k and girth(h) 6). Claim. If B = B 1 B d are the classes of H, then H = G [ A, i I B i] is C4 -free. Proof. If not... The intersection pattern J of e 1 e 2 contains {i 2, i 4 } I. Hence {i 2, i 4 } Γ, a contradiction!

44 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees.

45 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

46 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

47 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

48 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

49 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

50 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

51 20/34 Alternative Proof for g = 6 2 (Γ) is large = vertex b B, X b N G (b), Y b B \ {b} such that G[X b, Y b ] has large degrees. Proof of 2 :

52 21/34 Alternative Proof for g = 6 Proof of Theorem. If 1 holds for G, we are done! Otherwise 2 holds...

53 21/34 Alternative Proof for g = 6 Proof of Theorem. If 1 holds for G, we are done! Otherwise 2 holds...

54 21/34 Alternative Proof for g = 6 Proof of Theorem. If 1 holds for G[X b0, Y b0 ], we are done! Otherwise 2 holds...

55 21/34 Alternative Proof for g = 6 Proof of Theorem. If 1 holds for G[X b0, Y b0 ], we are done! Otherwise 2 holds...

56 21/34 Alternative Proof for g = 6 Proof of Theorem. If 1 holds for G[X b1, Y b1 ], we are done! Otherwise 2 holds...

57 21/34 Alternative Proof for g = 6 Proof of Theorem.

58 21/34 Alternative Proof for g = 6 Proof of Theorem. Found a complete bipartite subgraph G[{b 0,..., b m }, X bm ].

59 22/34 Question Is there a hypergraph lemma similar to Füredi s that could be used to prove the general case of the conjecture? More modestly, is there a hypergraph lemma that would allow us to prove the g = 8 case?

60 23/34 Lemma (Dellamonica R, 2011) For all k and g there exist c and d 0 such that the following holds. In any C 4 -free bipartite graph G = (A, B; E) satisfying d max { c log log (G), d 0 }, there exists H G such that d(h) k and girth(h) g.

61 24/34 Using the result of Kühn Osthus (the numerical bounds in DKMR are inferior), yields: Theorem For all k and g there exist α, β, and d 0 such that for any graph G with d(g) max { α ( log log (G) ) β, d0 } there exists H G such that d(h) k and girth(h) g.

62 24/34 Using the result of Kühn Osthus (the numerical bounds in DKMR are inferior), yields: Theorem For all k and g there exist α, β, and d 0 such that for any graph G with d(g) max { α ( log log (G) ) β, d0 } there exists H G such that d(h) k and girth(h) g. Recall: Pyber-R-Szemerédi yields same conclusion under stronger assumption d c log (G).

63 25/34 Proof of our lemma Assume wlog that G = (A, B; E) is such deg G (v) = d for all v A and B A. Take ϱ = e α k,g d, ε > 0, and partition B = B 0 B 1 B t, where B 0 = { v B : deg G (v) ϱ }, and B j = { v B : ϱ (1+ε)j 1 < deg G (v) ϱ (1+ε)j }, for j = 1,..., t.

64 25/34 Proof of our lemma Assume wlog that G = (A, B; E) is such deg G (v) = d for all v A and B A. Take ϱ = e α k,g d, ε > 0, and partition B = B 0 B 1 B t, where B 0 = { v B : deg G (v) ϱ }, and B j = { v B : ϱ (1+ε)j 1 < deg G (v) ϱ (1+ε)j }, for j = 1,..., t. t = O ( log log (G) ). For appropriate choice of c = c(k, g), (G) e ecd = t < d 8k.

65 26/34 Proof of our lemma

66 26/34 Proof of our lemma Easy case: e G (A, B 0 ) e(g) 2. Take G = G[A B 0 ] and note that (G ) ϱ e c d(g ). Pyber R Szemerédi s result implies that G contains the graph we are looking for.

67 27/34 Proof of our lemma Otherwise: by averaging, there exists j [t] such that t i=1 e G (A, B j ) e G (A, B i ) t e(g) d A 2t 2 d 8k = e(g) e G (A, B 0 ) t = 4k A.

68 27/34 Proof of our lemma Otherwise: by averaging, there exists j [t] such that { } e G (A, B j ) max 4k A, ϱ (1+ε)j 1 B j.

69 27/34 Proof of our lemma Otherwise: by averaging, there exists j [t] such that { } e G (A, B j ) max 4k A, ϱ (1+ε)j 1 B j. 1 Delete all vertices in B \ B j. 2 Sequentially delete vertices from A with degree < k and vertices of B j with degree < D/4, where D = ϱ (1+ε)j 1.

70 27/34 Proof of our lemma Otherwise: by averaging, there exists j [t] such that { } e G (A, B j ) max 4k A, ϱ (1+ε)j 1 B j. 1 Delete all vertices in B \ B j. 2 Sequentially delete vertices from A with degree < k and vertices of B j with degree < D/4, where D = ϱ (1+ε)j 1. Note: deleted at most k A + D 4 B j < e G (A, B j ) 2 edges.

71 28/34 Proof of our lemma Result: non-empty graph G with classes A and B = B j such that degrees in A are between k and d; degrees in B are between D/4 and D (1+ε) = ϱ (1+ε)j.

72 29/34 Back to hypergraphs Let H be a hypergraph with vertex set B and edges { NG (v) : v A }. Obs. 1: G is C 4 -free = e f 1 for all e, f H (i.e., H is a linear hypergraph).

73 29/34 Back to hypergraphs Let H be a hypergraph with vertex set B and edges { NG (v) : v A }. Obs. 1: G is C 4 -free = e f 1 for all e, f H (i.e., H is a linear hypergraph). Obs. 2: a cycle of length 2l in G corresponds to a cycle of length l in H: i.e., cycle (v 0, v 1,..., v 2l 1 ) corresponds to (e 0, e 1,..., e l 1 ), where e i = N G (v 2i ) H.

74 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l.

75 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... # H

76 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e 0... # H d 2

77 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e and e 1 w 1... # H d 2 D 1+ε

78 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e and e 1 w and w 2 e 1... # H d 2 D 1+ε d

79 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e and e 1 w and w 2 e # H d 2 D 1+ε d

80 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e and e 1 w and w 2 e and w l 1 e l 2... # H d l D (1+ε)(l 2)

81 30/34 Cycles in linear hypergraphs Recall: δ(h) D/4 and (H) D 1+ε. Let s count cycles of length l. 1 Pick an edge e 0 H... 2 and w 0 w 1 e and e 1 w and w 2 e and w l 1 e l at most one possible edge e l 1 v 0, v 2l 1. # H d l D (1+ε)(l 2) 1.

82 31/34 Cycles in linear hypergraphs Result: Number of cycles of length l is N l H d l D (1+ε)(l 2).

83 31/34 Cycles in linear hypergraphs Result: Number of cycles of length l is N l H d l D (1+ε)(l 2). Randomly select edges of H with probability p. How many cycles survive? In expectation, at most p l N l (p H ) p l 1 d l D (1+ε)(l 2).

84 31/34 Cycles in linear hypergraphs Result: Number of cycles of length l is N l H d l D (1+ε)(l 2). Randomly select edges of H with probability p. How many cycles survive? In expectation, at most p l N l (p H ) p l 1 d l D (1+ε)(l 2). }{{} 1 for small p

85 31/34 Cycles in linear hypergraphs Result: Number of cycles of length l is N l H d l D (1+ε)(l 2). Randomly select edges of H with probability p. How many cycles survive? In expectation, at most p l N l (p H ) p l 1 d l D (1+ε)(l 2). }{{} 1 for small p Linearity of expectation: p = D ε 1, ε = 1 2g = # of surviving edges is much larger than # of surviving cycles of length l < g/2.

86 31/34 Cycles in linear hypergraphs Result: Number of cycles of length l is N l H d l D (1+ε)(l 2). Randomly select edges of H with probability p. How many cycles survive? In expectation, at most p l N l (p H ) p l 1 d l D (1+ε)(l 2). }{{} 1 for small p Linearity of expectation: p = D ε 1, ε = 1 2g = # of surviving edges is much larger than # of surviving cycles of length l < g/2. Delete an edge for each cycle and destroy them all! Resulting hypergraph H has Ω ( D ε V (H) ) edges and no cycles of length < g/2.

87 32/34 Back to graphs Recall: edges of H correspond to vertices in A A (e H v A, e = N G (v)).

88 32/34 Back to graphs Recall: edges of H correspond to vertices in A A (e H v A, e = N G (v)). Let H be the induced subgraph G [ A V (H ) ]. girth(h) g because girth(h ) g/2;

89 32/34 Back to graphs Recall: edges of H correspond to vertices in A A (e H v A, e = N G (v)). Let H be the induced subgraph G [ A V (H ) ]. girth(h) g because girth(h ) g/2; by construction, deg H (v) k for all v A, thus d(h) = 2e(H) v(h) 2k A A + V (H ) = 2k H H + V (H ) > k.

90 32/34 Back to graphs Recall: edges of H correspond to vertices in A A (e H v A, e = N G (v)). Let H be the induced subgraph G [ A V (H ) ]. girth(h) g because girth(h ) g/2; by construction, deg H (v) k for all v A, thus d(h) = 2e(H) v(h) 2k A A + V (H ) = 2k H H + V (H ) > k.

91 32/34 Back to graphs Recall: edges of H correspond to vertices in A A (e H v A, e = N G (v)). Let H be the induced subgraph G [ A V (H ) ]. girth(h) g because girth(h ) g/2; by construction, deg H (v) k for all v A, thus d(h) = 2e(H) v(h) 2k A A + V (H ) = 2k H H + V (H ) > k. Hence, H is the desired subgraph!

92 33/34 Open questions The conjectures of Thomassen and Erdős Hajnal remain open. Is there a hypergraph lemma similar to Füredi s that can be used to prove Thomassen s conjecture for, say, g = 8? Can one extend the gap between d(g) and (G) for which we can establish that the conjecture is true? Hypergraph version of Thomassen s conjecture: any degree-gap result in hypergraphs extends to graphs from our proof. Here the degree gap is polynomial (H) δ(h) 1+ε.

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