Hamilton cycles in directed graphs

Transcription

1 Unversty of Brmngham, School of Mathematcs January 22nd, 2010 Jont work wth Danela Kühn and Deryk Osthus (Unversty of Brmngham)

2 Theorem (Drac, 1952) Graph G of order n 3 and δ(g) n/2 = G Hamltonan. Theorem (Ghoula-Hour, 1966) Dgraph G of order n 2 wth δ + (G), δ (G) n/2 = G Hamltonan.

3 Theorem (Chvátal, 1972) Let G be a graph wth degree sequence d 1 d n. G has a Hamlton cycle f d + 1 or d n n < n/2. The bound on the degrees n Chvátal s theorem s best possble. degree sequence: k,..., k, n k 1,..., n k 1, n 1,..., n 1 }{{}}{{}}{{} k tmes n 2k tmes k tmes

4 Theorem (Chvátal, 1972) Let G be a graph wth degree sequence d 1 d n. G has a Hamlton cycle f d + 1 or d n n < n/2. The bound on the degrees n Chvátal s theorem s best possble. complete k complete bpartte n k degree sequence: k,..., k, n k 1,..., n k 1, n 1,..., n 1 }{{}}{{}}{{} k tmes n 2k tmes k tmes

5 Theorem (Chvátal, 1972) Let G be a graph wth degree sequence d 1 d n. G has a Hamlton cycle f d + 1 or d n n < n/2. The bound on the degrees n Chvátal s theorem s best possble. complete k complete bpartte n k degree sequence: k,..., k, n k 1,..., n k 1, n 1,..., n 1 }{{}}{{}}{{} k tmes n 2k tmes k tmes

6 Nash-Wllams rased the queston of a dgraph analogue of Chvátal s theorem. Conjecture (Nash-Wllams, 1975) G strongly connected dgraph whose out- and ndegree sequences d 1 + d n + and d1 d n satsfy () d or d n n < n/2, () d + 1 or d + n n < n/2. Then G contans a Hamlton cycle. If true, the conjecture s much stronger than Ghoula-Hour s theorem.

7 If the Nash-Wllams conjecture s true then t s best possble. N-W conjecture () d or d n n () d + 1 or d + n n K = n k 2 and K = k 1 outdegree sequence: k 1,..., k 1, k, k, n 1,..., n 1 }{{}}{{} k 1 tmes n k 1 tmes ndegree sequence: n k 2,..., n k 2, n k 1, n k 1, n 1,..., n 1 }{{}}{{} n k 2 tmes k tmes

8 If the Nash-Wllams conjecture s true then t s best possble. N-W conjecture () d or d n n () d + 1 or d + n n K w u v K K = n k 2 and K = k 1 outdegree sequence: k 1,..., k 1, k, k, n 1,..., n 1 }{{}}{{} k 1 tmes n k 1 tmes ndegree sequence: n k 2,..., n k 2, n k 1, n k 1, n 1,..., n 1 }{{}}{{} n k 2 tmes k tmes

9 If the Nash-Wllams conjecture s true then t s best possble. N-W conjecture () d or d n n () d + 1 or d + n n K w u v K K = n k 2 and K = k 1 outdegree sequence: k 1,..., k 1, k, k, n 1,..., n 1 }{{}}{{} k 1 tmes n k 1 tmes ndegree sequence: n k 2,..., n k 2, n k 1, n k 1, n 1,..., n 1 }{{}}{{} n k 2 tmes k tmes

10 Theorem (Kühn, Osthus, T.) η > 0 n 0 = n 0 (η) s.t. f G s a dgraph on n n 0 vertces s.t. d + + ηn or d n ηn n < n/2, d + ηn or d + n ηn n < n/2, then G contans a Hamlton cycle. Corollary The condtons n the above theorem mply G s pancyclc. That s, G contans a cycle of length 2 G.

11 The followng result s an mmedate corollary of Chvátal s theorem. Theorem (Pósa, 1962) Let G be a graph of order n 3 wth degree sequence d 1 d n. G has a Hamlton cycle f d + 1 < (n 1)/2 and f addtonally d n/2 n/2 when n s odd. Pósa s theorem s much stronger than Drac s theorem.

12 The followng conjecture s a dgraph analogue of Pósa s theorem. Conjecture (Nash-Wllams, 1968) Let G be a dgraph on n 3 vertces s.t. d +, d + 1 < (n 1)/2 and s.t. d + n/2, d n/2 n/2 when n s odd. Then G contans a Hamlton cycle. If true, ths conjecture s much stronger than Ghoula-Hour s theorem.

13 Theorem (Kühn, Osthus, T.) η > 0 n 0 = n 0 (η) s.t. f G s a dgraph on n n 0 vertces s.t. d + + ηn or d n ηn n < n/2, d + ηn or d + n ηn n < n/2, then G contans a Hamlton cycle. Ths theorem mples an approxmate verson of the second Nash-Wllams conjecture. Corollary η > 0 n 0 = n 0 (η) s.t. every dgraph G on n n 0 vertces wth d +, d + ηn < n/2 contans a Hamlton cycle.

14 Theorem (Kühn, Osthus, T.) η > 0 n 0 = n 0 (η) s.t. f G s a dgraph on n n 0 vertces s.t. d + + ηn or d n ηn n < n/2, d + ηn or d + n ηn n < n/2, then G contans a Hamlton cycle. Ths theorem mples an approxmate verson of the second Nash-Wllams conjecture. Corollary η > 0 n 0 = n 0 (η) s.t. every dgraph G on n n 0 vertces wth d +, d + ηn < n/2 contans a Hamlton cycle.

15 Recent developments... Chrstofdes, Keevash, Kühn and Osthus gave a polynomal tme algorthm whch fnds a Hamlton cycle n those dgraphs consdered n our result. They also showed one can relax the condton n our result to d + mn{ + ηn, n/2} or d n ηn n < n/2, d mn{ + ηn, n/2} or d + n ηn n < n/2,.

16 Hamlton cycles n orented graphs Theorem (Keevash, Kühn, Osthus, 2009) n 0 s.t. every orented graph G on n n 0 vertces wth contans a Hamlton cycle. δ + (G), δ (G) 3n 4 8 Queston Can we strengthen ths theorem n the same way as Pósa s theorem strengthens Drac s theorem?

17 Hamlton cycles n orented graphs Theorem (Keevash, Kühn, Osthus, 2009) n 0 s.t. every orented graph G on n n 0 vertces wth contans a Hamlton cycle. δ + (G), δ (G) 3n 4 8 Queston Can we strengthen ths theorem n the same way as Pósa s theorem strengthens Drac s theorem?

18 Let 0 < α < 3/8, G = n suffcently large, c = c(α) constant. A B C D n/4 n/8 n/8 1 n/4 + 1 E n/4 Both n- and outdegree sequences domnate αn,..., αn, 3n/8,..., 3n/8 }{{} c tmes

19 Let 0 < α < 3/8, G = n suffcently large, c = c(α) constant. A B C D n/4 n/8 n/8 1 n/4 + 1 E n/4 Both n- and outdegree sequences domnate αn,..., αn, 3n/8,..., 3n/8 }{{} c tmes

20 Robustly expandng dgraphs To prove the approxmate verson of the Nash-Wllams conjecture we n fact showed that... Robustly expandng dgraphs of large enough mnmum degree are Hamltonan Ths mples approxmate verson of the theorem of Keevash, Kühn and Osthus. Used n proof of approxmate Sumner s Unversal Tournament conjecture by Kühn, Mycroft and Osthus.

21 Robustly expandng dgraphs To prove the approxmate verson of the Nash-Wllams conjecture we n fact showed that... Robustly expandng dgraphs of large enough mnmum degree are Hamltonan Ths mples approxmate verson of the theorem of Keevash, Kühn and Osthus. Used n proof of approxmate Sumner s Unversal Tournament conjecture by Kühn, Mycroft and Osthus.

22 Hamlton decompostons Hamlton decomposton of a graph or dgraph G: set of edge-dsjont Hamlton cycles coverng E(G) Theorem (Waleck 1892) K n has a Hamlton decomposton n odd

23 Hamlton decompostons Hamlton decomposton of a graph or dgraph G: set of edge-dsjont Hamlton cycles coverng E(G) Theorem (Waleck 1892) K n has a Hamlton decomposton n odd

24 Hamlton decompostons n dgraphs Theorem (Tllson 1980) Complete dgraph on n vertces has Hamlton decomposton n 4, 6. Tournament: orentaton of a complete graph Tournament on n vertces s regular f every vertex has equal n- and outdegree (.e. (n-1)/2)

25 Hamlton decompostons n dgraphs Theorem (Tllson 1980) Complete dgraph on n vertces has Hamlton decomposton n 4, 6. Tournament: orentaton of a complete graph Tournament on n vertces s regular f every vertex has equal n- and outdegree (.e. (n-1)/2)

26 Hamlton decompostons n dgraphs Theorem (Tllson 1980) Complete dgraph on n vertces has Hamlton decomposton n 4, 6. Tournament: orentaton of a complete graph Tournament on n vertces s regular f every vertex has equal n- and outdegree (.e. (n-1)/2)

27 Hamlton decompostons n dgraphs Theorem (Tllson 1980) Complete dgraph on n vertces has Hamlton decomposton n 4, 6. Tournament: orentaton of a complete graph Tournament on n vertces s regular f every vertex has equal n- and outdegree (.e. (n-1)/2)

28 Conjecture (Kelly 1968) All regular tournaments have Hamlton decompostons. There have been several partal results n ths drecton. Keevash, Kühn and Osthus: G orented graph δ + (G), δ (G) (3 G 4)/8 = H.C. So regular tournament G contans G /8 edge-dsjont Hamlton cycles

29 Conjecture (Kelly 1968) All regular tournaments have Hamlton decompostons. There have been several partal results n ths drecton. Keevash, Kühn and Osthus: G orented graph δ + (G), δ (G) (3 G 4)/8 = H.C. So regular tournament G contans G /8 edge-dsjont Hamlton cycles

30 Conjecture (Thomassen 1982) Suppose G regular tournament on n vertces and A E(G) s.t A < (n 1)/2. Then G A contans a Hamlton cycle. Theorem (Kühn, Osthus, T.) Conjecture true for large n

31 Theorem (Kühn, Osthus, T.) η > 0 n 0 s.t all regular tournaments on n n 0 vertces contan (1/2 η)n edge-dsjont Hamlton cycles. In fact, result holds for almost regular tournaments.

32 Naïve approach to theorem Remove a γn-regular orented spannng subgraph H from G (γ 1). Decompose rest of G nto 1-factors F 1,..., F s. Use edges from H to pece together each F nto Hamlton cycles. Need F to contan few cycles (a result of Freze and Krvelevch mples ths). If H quas-random could use t to merge cycles usng method of rotaton-extenson. Problem: can t necessarly fnd such H. But ths approach s a useful startng pont.

33 Naïve approach to theorem Remove a γn-regular orented spannng subgraph H from G (γ 1). Decompose rest of G nto 1-factors F 1,..., F s. Use edges from H to pece together each F nto Hamlton cycles. Need F to contan few cycles (a result of Freze and Krvelevch mples ths). If H quas-random could use t to merge cycles usng method of rotaton-extenson. Problem: can t necessarly fnd such H. But ths approach s a useful startng pont.

34 Naïve approach to theorem Remove a γn-regular orented spannng subgraph H from G (γ 1). Decompose rest of G nto 1-factors F 1,..., F s. Use edges from H to pece together each F nto Hamlton cycles. Need F to contan few cycles (a result of Freze and Krvelevch mples ths). If H quas-random could use t to merge cycles usng method of rotaton-extenson. Problem: can t necessarly fnd such H. But ths approach s a useful startng pont.

35 Open problems Kelly s conjecture! Theorem (Kühn, Osthus, T.) Almost regular orented graphs G wth δ + (G), δ (G) (3/8 + o(1)) G can be almost decomposed nto edge-dsjont Hamlton cycles. Queston What mnmum degree ensures a regular orented graph has a Hamlton decomposton?

36 Open problems Kelly s conjecture! Theorem (Kühn, Osthus, T.) Almost regular orented graphs G wth δ + (G), δ (G) (3/8 + o(1)) G can be almost decomposed nto edge-dsjont Hamlton cycles. Queston What mnmum degree ensures a regular orented graph has a Hamlton decomposton?

37 Open problems cont. Conjecture (Jackson) All regular bpartte tournaments have Hamlton decompostons. Almost regular bpartte tournaments may not even contan a Hamlton cycle. No Hamlton cycle

38 Open problems cont. Conjecture (Jackson) All regular bpartte tournaments have Hamlton decompostons. Almost regular bpartte tournaments may not even contan a Hamlton cycle. m + 1 m m m 1 No Hamlton cycle

39 Open problems cont. Conjecture (Jackson) All regular bpartte tournaments have Hamlton decompostons. Almost regular bpartte tournaments may not even contan a Hamlton cycle. m + 1 m m m 1 No Hamlton cycle

40 Open problems cont. Problem of Erdős: Do almost all tournaments T have mn{δ + (T ), δ (T )} edge-dsjont Hamlton cycles? Conjecture (Bang-Jansen, Yeo 2004) Every k-edge-connected tournament has a decomposton nto k spannng strong dgraphs.

41 Open problems cont. Problem of Erdős: Do almost all tournaments T have mn{δ + (T ), δ (T )} edge-dsjont Hamlton cycles? Conjecture (Bang-Jansen, Yeo 2004) Every k-edge-connected tournament has a decomposton nto k spannng strong dgraphs.