Finding many edge-disjoint Hamiltonian cycles in dense graphs

Transcription

1 Finding many edge-disjoint Hamiltonian cycles in dense graphs Stephen G. Hartke Department of Mathematics University of Nebraska Lincoln shartke2 Joint work with Tyler Seacrest

2 Finding Edge-Disj. Subgraphs in Dense Graphs Ques. How many edge-disjoint 1-factors can we find in a dense graph G with minimum degree δ n/2? How many edge-disjoint Hamiltonian cyles?

3 Finding Edge-Disj. Subgraphs in Dense Graphs Ques. How many edge-disjoint 1-factors can we find in a dense graph G with minimum degree δ n/2? How many edge-disjoint Hamiltonian cyles? Thm. [Katerinis 1985; Egawa and Enomoto 1989] If δ n/2, then G contains a k-factor for all k (n + 5)/4. This is best possible. Thus, roughly n/ 4 edge-disjoint 1-factors and n/8 edge-disjoint Ham cycles are the best we can hope for.

4 Finding Edge-Disj. Subgraphs in Dense Graphs Ques. How many edge-disjoint 1-factors can we find in a dense graph G with minimum degree δ n/2? How many edge-disjoint Hamiltonian cyles? Thm. [Katerinis 1985; Egawa and Enomoto 1989] If δ n/2, then G contains a k-factor for all k (n + 5)/4. This is best possible. Thus, roughly n/ 4 edge-disjoint 1-factors and n/8 edge-disjoint Ham cycles are the best we can hope for. Thm. [Nash-Williams, 1971] If δ n/2, then G contains at least 5n/224 edge-disjoint Hamiltonian cycles. This gives roughly n/ 46 edge-disjoint Hamiltonian cycles, which gives n/ 23 edge-disjoint 1-factors.

5 Results Thm. [Christofides, Kühn, Osthus, 2011+] For every ε > 0, if n is sufficiently large and δ (1/2 + ε)n, then G contains n/ 8 Hamiltonian cycles.

6 Results Thm. [Christofides, Kühn, Osthus, 2011+] For every ε > 0, if n is sufficiently large and δ (1/2 + ε)n, then G contains n/ 8 Hamiltonian cycles. Under the hypotheses, this gives n/ 4 edge-disjoint 1-factors. CKO s proof uses the Regularity Lemma.

7 Results Thm. [Christofides, Kühn, Osthus, 2011+] For every ε > 0, if n is sufficiently large and δ (1/2 + ε)n, then G contains n/ 8 Hamiltonian cycles. Under the hypotheses, this gives n/ 4 edge-disjoint 1-factors. CKO s proof uses the Regularity Lemma. Thm. [H, Seacrest 2011+] Let G have min deg δ n/2 + O(n 3/4 ln(n)). Then G contains n/8 O(n 7/8 ln(n)) edge-disjoint Ham cycles. Our proof avoids the Regularity Lemma and hence the constants are much smaller.

8 Results Thm. [Christofides, Kühn, Osthus, 2011+] For every ε > 0, if n is sufficiently large and δ (1/2 + ε)n, then G contains n/ 8 Hamiltonian cycles. Under the hypotheses, this gives n/ 4 edge-disjoint 1-factors. CKO s proof uses the Regularity Lemma. Thm. [H, Seacrest 2011+] Let G have min deg δ n/2 + O(n 3/4 ln(n)). Then G contains n/8 O(n 7/8 ln(n)) edge-disjoint Ham cycles. Our proof avoids the Regularity Lemma and hence the constants are much smaller. (All of our theorems also describe when δ > n/2.)

9 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H with at least half the edges of G.

10 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H with at least half the edges of G. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part.

11 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H with at least half the edges of G. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part. Repeat; the number of edges in H strictly increases.

12 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) deg G ( )/2 for all vertices. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part. Repeat; the number of edges in H strictly increases.

13 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) deg G ( )/2 for all vertices. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part. Repeat; the number of edges in H strictly increases. Prop. Every graph G has a bipartite subgraph H with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[ E(H) ] E(G) /2.

14 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) deg G ( )/2 for all vertices. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part. Repeat; the number of edges in H strictly increases. Prop. Every graph G has a bipartite subgraph H with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[ E(H) ] E(G) /2. The same analysis works with a random balanced partition.

15 Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) deg G ( )/2 for all vertices. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( )/2, then push to the other part. Repeat; the number of edges in H strictly increases. Prop. Every graph G has a balanced bipartite subgraph H with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[ E(H) ] E(G) /2. The same analysis works with a random balanced partition.

16 Large Bipartite Subgraphs Can both properties be simultaneously obtained?

17 Large Bipartite Subgraphs Can both properties be simultaneously obtained? A bisection is a balanced spanning bipartite subgraph. Conj. [Bollobás Scott 2002] Every graph G with an even number of vertices has a bisection H with deg H ( ) deg G ( )/2 for all.

18 Sharpness The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with deg H ( ) deg G ( )/2 + 1 for all. Let k < n/2. K k n k K k (n k)k 1

19 Sharpness The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with deg H ( ) deg G ( )/2 + 1 for all. Let k < n/2. K k K k (n k)k 1 n k For any partition, both partite sets have vertices from the independent set.

20 Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices, deg H ( ) 1 2 deg G( ) deg G ( ) ln(n). [similar result by A. Bush 2009]

21 Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices, deg H ( ) 1 2 deg G( ) deg G ( ) ln(n). [similar result by A. Bush 2009] Proof Outline. Arbitrarily pair the vertices. Randomly split each pair between the two partite sets.

22 Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices, deg H ( ) 1 2 deg G( ) deg G ( ) ln(n). [similar result by A. Bush 2009] Proof Outline. Arbitrarily pair the vertices. Randomly split each pair between the two partite sets. Bound the probability of a bad vertex using Chernoff bounds. Combine using the union sum bound.

23 Larger Partitions Thm. Let G be a graph on n vertices, where n = pq for p > 1. Then there exists a partition of G into q parts of size p such that every vertex has at least deg( )/q deg( ) ln(n) neighbors in each part.

24 Larger Partitions Thm. Let G be a graph on n vertices, where n = pq for p > 1. Then there exists a partition of G into q parts of size p such that every vertex has at least deg( )/q deg( ) ln(n) neighbors in each part. An upper bound can be obtained following a construction of Doerr and Srivastav 2003 from discrepancy theory. It is based on a construction of Spencer 1985 using Hadamard matrices. Thm. For infinitely many n, there exists a graph G on n vertices such than any partition of G into q parts contains a part P and vertex such that has less than deg( )/q 1 n/q 3 3 neighbors in P.

25 Finding k-factors in Bipartite Graphs Thm. [Csaba 2007] Let G be balanced bipart. graph on 2p vertices with min deg δ p 2. Let α = δ + p(2δ p). 2 Then G has an α -regular spanning subgraph.

26 Finding k-factors in Bipartite Graphs Thm. [Csaba 2007] Let G be balanced bipart. graph on 2p vertices with min deg δ p 2. Let α = δ + p(2δ p). 2 Then G has an α -regular spanning subgraph. Thm. If G has min deg δ n/2 + 2 n/2 ln(n), then G contains n/ 8 edge-disjoint 1-factors.

27 Finding k-factors in Bipartite Graphs Thm. [Csaba 2007] Let G be balanced bipart. graph on 2p vertices with min deg δ p 2. Let α = δ + p(2δ p). 2 Then G has an α -regular spanning subgraph. Thm. If G has min deg δ n/2 + 2 n/2 ln(n), then G contains n/ 8 edge-disjoint 1-factors. Proof. G has bisection H with min deg δ/2 δ ln(n). Apply Csaba to get regular subgraph in H.

28 Edge-Disjoint 1-Factors Thm. If G has n verts, n = pq, q even, with δ(g) n/2 + q p ln(n), then G has (n p)/4 edge-disjoint 1-factors.

29 Edge-Disjoint 1-Factors Thm. If G has n verts, n = pq, q even, with δ(g) n/2 + q p ln(n), then G has (n p)/4 edge-disjoint 1-factors. Cor. If G has n verts, n even and square, with δ(g) n/2 + (n ln n) 3 4, then G has n/4 n/4 edge-disjoint 1-factors.

30 Edge-Disjoint 1-Factors Proof. Obtain q parts of size p.

31 Edge-Disjoint 1-Factors Proof. Obtain q parts of size p. Between the parts are bisections of large min deg.

32 Edge-Disjoint 1-Factors Proof. Obtain q parts of size p. Between the parts are bisections of large min deg. Apply Csaba s thm to split into matchings.

33 Edge-Disjoint 1-Factors Proof. Obtain q parts of size p. Between the parts are bisections of large min deg. Apply Csaba s thm to split into matchings. Combine into 1-factors by blowing up 1-factors of K q.

34 General Case: 1-Factors A similar result holds when n is not (close to) square. Lem. For any positive integer n, there exists an integer p n such that if q = n/p, then q is even, p n < 2n 1/4 + 4, n q < 2n 1/4 + 4, and n pq < 4 2n 1/ In particular, a set of size n can be partitioned into parts of size p and p + 1 such that there are at most 2 2n 1/4 + 5 parts of size p + 1.

35 General Case: 1-Factors Cor. Let G be a graph on n vertices, q < n and p = n/q. There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg( )/q min{deg( ), p + 1} ln(n + q) nbrs in each part.

36 General Case: 1-Factors Cor. Let G be a graph on n vertices, q < n and p = n/q. There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg( )/q min{deg( ), p + 1} ln(n + q) nbrs in each part. Thus we partition G into n parts of size p or p + 1. The number of big parts is small: 2 2n 1/4 + 5.

37 General Case: 1-Factors Cor. Let G be a graph on n vertices, q < n and p = n/q. There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg( )/q min{deg( ), p + 1} ln(n + q) nbrs in each part. Thus we partition G into n parts of size p or p + 1. The number of big parts is small: 2 2n 1/ How do we find 1-factors between parts of different size?

38 Generalization of Csaba s Theorem Lem. Let G be bipartite on X Y with X = p + 1 and Y = p such that vertices in X have min deg δ and vertices in Y have min deg δ + 1 for some δ p/2. Let k δ+ 2δp p 2 2 and let ƒ : V(G) N satisfy ƒ ( ) k for X, X = p + 1 Y = p ƒ ( ) = k for Y, and X ƒ ( ) = Y ƒ ( ). Then G has an ƒ-factor H. Moreover, H decomposes into matchings that saturate Y. k k k k k k k δ δ + 1

39 Blowing Up We only blow up 1-factors of K q that have no edges between parts of size p + 1.

40 Blowing Up We only blow up 1-factors of K q that have no edges between parts of size p + 1. Lem. Let q be even, and let A be a set in K q with A = 2. Then there exist q edge-disjoint 1-factors in K q with no edges inside A.

41 Putting It All Together Partition G into q n parts of size p and p + 1, p n, with 2 2n 1/4 + 5 parts of size p + 1; call these parts A. p + 1 p

42 Putting It All Together Partition G into q n parts of size p and p + 1, p n, with 2 2n 1/4 + 5 parts of size p + 1; call these parts A. There are q 8 2n 1/4 14 edge-disjoint 1-factors of K q that use no edges of A. p + 1 p

43 Putting It All Together Partition G into q n parts of size p and p + 1, p n, with 2 2n 1/4 + 5 parts of size p + 1; call these parts A. There are q 8 2n 1/4 14 edge-disjoint 1-factors of K q that use no edges of A. Blow up using Csaba s thm and generalization, obtaining n/4 O(n 7/8 ln(n)) edge-disjoint 1-factors. p + 1 p

44 Hamiltonian Cycles Thm. Let G have min deg δ n/2 + O(n 3/4 ln(n)). Then G contains n/8 O(n 7/8 ln(n)) edge-disjoint Ham cycles.

45 Hamiltonian Cycles Thm. Let G have min deg δ n/2 + O(n 3/4 ln(n)). Then G contains n/8 O(n 7/8 ln(n)) edge-disjoint Ham cycles. Proof sketch. We partition V(G) into parts of size p and p + 1. There are few parts of size p + 1 (call them set A). Make a set B of size n 3/8 of parts of size p, set C the rest.

46 Hamiltonian Cycles in K q Lem. Fix K q. Let A, B, C be a vertex partition as before. Then there are q 2 /2 2q 7/4 Hamiltonian cycles (not necessarily edge-disjoint) satisfying every edge of K q appears in at most q + 8q ln(q) cycles, no edge within A appears in a cycle, every edge within B appears in at most q cycles, every cycle has exactly one edge within B.

47 Blowing Up Hamiltonian Cycles Blow up each Ham cycle of K q using 1-factors between parts.

48 Blowing Up Hamiltonian Cycles Blow up each Ham cycle of K q using 1-factors between parts. Glue together using the one edge in B.

49 Open Questions Can the error term in the probabilistic lemma be improved? Can the ln n term be removed? Prove the Bollobás Scott conjecture. What can be proved when the minimum degree is not large? Other uses of our partition theorem?

50 Finding many edge-disjoint Hamiltonian cycles in dense graphs Stephen G. Hartke Department of Mathematics University of Nebraska Lincoln shartke2 Joint work with Tyler Seacrest