An asymptotic answer to a special case of an open conjecture of Bondy

Transcription

1 to a special case of an open conjecture of Bondy Peter Heinig Technische Universität München 30. Kolloquium über Kombinatorik Otto-von-Guericke-Universität Magdeburg 11. November 2011

2 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q

3 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q

4 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q Conjecture (A. Bondy 1979) For every d P Z, in every vertex-3-connected graph X with f 0 pxq ě 2d and δpxq ě d, the set of all circuits of length at least 2d 1 is a Z{2-generating system of Z 1 px; Z{2q.

5 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q Conjecture (A. Bondy 1979) For every d P Z, in every vertex-3-connected graph X with f 0 pxq ě 2d and δpxq ě d, the set of all circuits of length at least 2d 1 is a Z{2-generating system of Z 1 px; Z{2q. Theorem (H. 2011) If δpxq ě d is replaced by δpxq ě p1 ` γqd for an arbitrary γ ą 0, and if f 0 pxq is sufficiently large, then in the case of f 0 pxq 2d Bondy s conjecture is true.

6 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q Conjecture (A. Bondy 1979) For every d P Z, in every vertex-3-connected graph X with f 0 pxq ě 2d and δpxq ě d, the set of all circuits of length at least 2d 1 is a Z{2-generating system of Z 1 px; Z{2q. Theorem (H. 2011) If δpxq ě d is replaced by δpxq ě p1 ` γqd for an arbitrary γ ą 0, and if f 0 pxq is sufficiently large, then in the case of f 0 pxq 2d Bondy s conjecture is true. Moreover, in the case of f 0 pxq 2d, the circuits of length 2d in X (Hamilton circuits) generate a codimension-1-subspace of Z 1 px; Z{2q.

7 A conjecture of Bondy f 0 pxq : number of vertices of graph X δpxq : minimum vertex degree of graph X Z 1 px; Z{2q : ker`b : C 1 px; Z{2q ÝÑ C 0 px; Z{2q Conjecture (A. Bondy 1979) For every d P Z, in every vertex-3-connected graph X with f 0 pxq ě 2d and δpxq ě d, the set of all circuits of length at least 2d 1 is a Z{2-generating system of Z 1 px; Z{2q. Theorem (H. 2011) If δpxq ě d is replaced by δpxq ě p1 ` γqd for an arbitrary γ ą 0, and if f 0 pxq is sufficiently large, then in the case of f 0 pxq 2d Bondy s conjecture is true. If f 0 pxq 2d ` 1, then of the three circuit lengths tf 0 pxq 2, f 0 pxq 1, f 0 pxqu allowed by Bondy s conjecture, f 0 pxq alone is enough (Hamilton circuits).

8 A practically-sized statement large enough not to be provable by brute-force search

9 A practically-sized statement large enough not to be provable by brute-force search Theorem (H. 2011) For every graph X with vertices and minimum vertex degree at least , the Hamilton circuits of X are a Z{2-generating system of Z 1 px; Z{2q.

10 A practically-sized statement large enough not to be provable by brute-force search Theorem (H. 2011) For every graph X with vertices and minimum vertex degree at least , the Hamilton circuits of X are a Z{2-generating system of Z 1 px; Z{2q. Provable by combining the argumentation outlined in this talk with a very recent theorem of P. Châu, L. DeBiasio and H. A. Kierstead [Random Structures & Algorithms 39 (2011)].

11 A practically-sized statement large enough not to be provable by brute-force search Theorem (H. 2011) For every graph X with vertices and minimum vertex degree at least , the Hamilton circuits of X are a Z{2-generating system of Z 1 px; Z{2q. Provable by combining the argumentation outlined in this talk with a very recent theorem of P. Châu, L. DeBiasio and H. A. Kierstead [Random Structures & Algorithms 39 (2011)]. The explicit numbers here do not have any absolute meaning and are likely to be improved soon. The theorem on this slide is a snapshot of a rapidly evolving theory.

12 A practically-sized statement large enough not to be provable by brute-force search Theorem (H. 2011) For every graph X with vertices and minimum vertex degree at least , the Hamilton circuits of X are a Z{2-generating system of Z 1 px; Z{2q. Provable by combining the argumentation outlined in this talk with a very recent theorem of P. Châu, L. DeBiasio and H. A. Kierstead [Random Structures & Algorithms 39 (2011)]. The explicit numbers here do not have any absolute meaning and are likely to be improved soon. The theorem on this slide is a snapshot of a rapidly evolving theory. It seems likely that the hypothesis can be weakened to a minimum degree of , but this is not known.

13 The only other known sufficient condition The first known sufficient condition for a cycle space generated by Hamilton circuits was (let HpXq : t Hamilton circuits of X u): Theorem (B. Alspach, S. C. Locke, D. Witte 1990) If X is a connected Cayley graph on a finite abelian group G, and if either G is odd or X is bipartite, then Z 1 px; Z{2q xhpxqy Z{2.

14 The only other known sufficient condition The first known sufficient condition for a cycle space generated by Hamilton circuits was (let HpXq : t Hamilton circuits of X u): Theorem (B. Alspach, S. C. Locke, D. Witte 1990) If X is a connected Cayley graph on a finite abelian group G, and if either G is odd or X is bipartite, then Z 1 px; Z{2q xhpxqy Z{2. The result of this talk apparently is the first known degree-condition guaranteeing Z 1 px; Z{2q xhpxqy Z{2.

15 The only other known sufficient condition The first known sufficient condition for a cycle space generated by Hamilton circuits was (let HpXq : t Hamilton circuits of X u): Theorem (B. Alspach, S. C. Locke, D. Witte 1990) If X is a connected Cayley graph on a finite abelian group G, and if either G is odd or X is bipartite, then Z 1 px; Z{2q xhpxqy Z{2. The result of this talk apparently is the first known degree-condition guaranteeing Z 1 px; Z{2q xhpxqy Z{2. The graph underlying the following example is a 4-regular non-cayley graph having Z 1 px; Z{2q xhpxqy Z{2 :

16 Essence of the proof The set of graphs "X : X Hamilton-connected & Z 1 px; Z{2q xhpxqy Z{2 * is a monotone graph property.

17 A tool in the proof Theorem (J. Böttcher, M. Schacht, A. Taraz γ ą 0 : D β ą 0 : D n 0 graph X with f 0 pxq ě n 0 and δpxq ě p 1 2 ` γq f 0pXq graph Y with f 0 pyq f 0 pxq and pyq ď and bwpyq ď β f 0 pyq and χpyq ď 3 with small third colour class : D embedding Y ãñ X.

18 A tool in the proof Theorem (J. Böttcher, M. Schacht, A. Taraz γ ą 0 : D β ą 0 : D n 0 graph X with f 0 pxq ě n 0 and δpxq ě p 1 2 ` γq f 0pXq graph Y with f 0 pyq f 0 pxq and pyq ď and bwpyq ď β f 0 pyq and χpyq ď 3 with small third colour class : D embedding Y ãñ X. In particular:

19 A tool in the proof Theorem (J. Böttcher, M. Schacht, A. Taraz γ ą 0 : D β ą 0 : D n 0 graph X with f 0 pxq ě n 0 and δpxq ě p 1 2 ` γq f 0pXq graph Y with f 0 pyq f 0 pxq and pyq ď and bwpyq ď β f 0 pyq and χpyq ď 3 with small third colour class : D embedding Y ãñ X. In particular: spanningly X with f 0 pxq odd and δpxq ě p 1 2 ` γqf 0pXq

20 A tool in the proof Theorem (J. Böttcher, M. Schacht, A. Taraz γ ą 0 : D β ą 0 : D n 0 graph X with f 0 pxq ě n 0 and δpxq ě p 1 2 ` γq f 0pXq graph Y with f 0 pyq f 0 pxq and pyq ď and bwpyq ď β f 0 pyq and χpyq ď 3 with small third colour class : D embedding Y ãñ X. In particular: spanningly X with f 0 pxq odd and δpxq ě p 1 2 ` γqf 0pXq spanningly X with f 0 pxq even and δpxq ě p 1 2 ` γqf 0pXq

21 A tool in the proof Theorem (J. Böttcher, M. Schacht, A. Taraz γ ą 0 : D β ą 0 : D n 0 graph X with f 0 pxq ě n 0 and δpxq ě p 1 2 ` γq f 0pXq graph Y with f 0 pyq f 0 pxq and pyq ď and bwpyq ď β f 0 pyq and χpyq ď 3 with small third colour class : D embedding Y ãñ X. In particular: spanningly X with f 0 pxq odd and δpxq ě p 1 2 ` γqf 0pXq spanningly X with f 0 pxq even and δpxq ě p 1 2 ` γqf 0pXq

22 A tool in the proof

23 The Hamilton circuit basis used in the proof

24 The Hamilton circuit basis used in the proof

25 The Hamilton circuit basis used in the proof

26 The Hamilton circuit basis used in the proof

27 The Hamilton circuit basis used in the proof

28 The Hamilton circuit basis used in the proof

29 The Hamilton circuit basis used in the proof

30 The Hamilton circuit basis used in the proof

31 The Hamilton circuit basis used in the proof

32 work: Synthesis with recent asymptotic theorems on the structure of the set HpXq

33 work: Synthesis with recent asymptotic theorems on the structure of the set HpXq D. Christofides, D. Kühn, D. Osthus 2009: Asymptotic theorems approximating the following open conjectures: Every d-regular graph X with d ě f0pxq 1 2 realizes the obvious upper bound of t 1 2du for the number of mutually edge-disjoint Hamilton circuits.

34 work: Synthesis with recent asymptotic theorems on the structure of the set HpXq D. Christofides, D. Kühn, D. Osthus 2009: Asymptotic theorems approximating the following open conjectures: Every d-regular graph X with d ě f0pxq 1 2 realizes the obvious upper bound of t 1 2du for the number of mutually edge-disjoint Hamilton circuits. Every graph X with δpxq ě 1 2 f 0pXq contains at least 1 8 f 0pXq edge-disjoint Hamilton circuits.

35 work: Synthesis with recent asymptotic theorems on the structure of the set HpXq D. Christofides, D. Kühn, D. Osthus 2009: Asymptotic theorems approximating the following open conjectures: Every d-regular graph X with d ě f0pxq 1 2 realizes the obvious upper bound of t 1 2du for the number of mutually edge-disjoint Hamilton circuits. Every graph X with δpxq ě 1 2 f 0pXq contains at least 1 8 f 0pXq edge-disjoint Hamilton circuits. F. Knox, D. Kühn, D. Osthus 2011: A theorem proving a large part of the following open conjecture: For any p n, an Erdős Rényi random graph G n,pn a.a.s. realizes the obvious upper bound t 1 2 δpg n,p n qu for the number of mutually edge-disjoint Hamilton circuits.

36 work: Role of the set HpXq vis-à-vis the group Z 1 px; Zq

37 work: Role of the set HpXq vis-à-vis the group Z 1 px; Zq The group Z 1 px; Zq is traditionally studied within the context of integral homology of simplicial complexes.

38 work: Role of the set HpXq vis-à-vis the group Z 1 px; Zq The group Z 1 px; Zq is traditionally studied within the context of integral homology of simplicial complexes. Over Z, tangible new obstacles arise.

39 work: Role of the set HpXq vis-à-vis the group Z 1 px; Zq The group Z 1 px; Zq is traditionally studied within the context of integral homology of simplicial complexes. Over Z, tangible new obstacles arise. Computer experiments show that cannot possibly serve to prove Z 1 px; Zq xhpxqy Z in the same way as it does when proving Z 1 px; Z{2q xhpxqy Z{2.

40 work: Role of the set HpXq vis-à-vis the group Z 1 px; Zq The group Z 1 px; Zq is traditionally studied within the context of integral homology of simplicial complexes. Over Z, tangible new obstacles arise. Computer experiments show that cannot possibly serve to prove Z 1 px; Zq xhpxqy Z in the same way as it does when proving Z 1 px; Z{2q xhpxqy Z{2. One hits upon large (and odd) torsion. If Pr b r : r, then Z 1 ppr b r ; Zq{xHpPr b r qy Z Z{p2r 1qZ.