Hamilton Circuits and Dominating Circuits in Regular Matroids S. McGuinness Thompson Rivers University June 1, 2012 S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 1 / 22
A Brief History Theorem (Dirac, 1952) Let G be a simple graph G of order n 3. If each vertex has degree at least n 2, then G has a Hamilton cycle. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 2 / 22
A Brief History Theorem (Dirac, 1952) Let G be a simple graph G of order n 3. If each vertex has degree at least n 2, then G has a Hamilton cycle. A cycle C is a graph G is dominating if every edge has at least one endvertex on C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 2 / 22
A Brief History Theorem (Dirac, 1952) Let G be a simple graph G of order n 3. If each vertex has degree at least n 2, then G has a Hamilton cycle. A cycle C is a graph G is dominating if every edge has at least one endvertex on C. Theorem (Nash-Williams, 1971) Let G be a simple 2-connected graph of order n 3. If d G (v) 1 3 (n + 2) for all vertices v, then any longest cycle in G is also a dominating cycle. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 2 / 22
Edge-Disjoint Hamilton Circuits This theorem was used to prove: Theorem (Nash-Williams, 1971) Every simple graph on n vertices of minimum degree at least n 2 least 5n 224 edge-disjoint hamilton cycles. contains at This theorem was recently extended. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 3 / 22
Edge-Disjoint Hamilton Circuits This theorem was used to prove: Theorem (Nash-Williams, 1971) Every simple graph on n vertices of minimum degree at least n 2 least 5n 224 edge-disjoint hamilton cycles. contains at This theorem was recently extended. Theorem (Christofides, Kühn, Osthus, 2011) For every α > 0 there is an integer n 0 so that every graph on n n 0 vertices of minimum degree at least ( 1 2 + α)n contains at least n 8 edge-disjoint Hamilton cycles. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 3 / 22
Regular Matroids A matroid M is regular if for every field F, M can be represented by some matrix over F S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 4 / 22
Regular Matroids A matroid M is regular if for every field F, M can be represented by some matrix over F Another definition: A matroid is regular if it is binary and has no F 7 - or F 7 -minor. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 4 / 22
The matroid F 7. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 5 / 22
Welsh s Conjecture A hamilton circuit in a matroid M is a circuit of size r(m) + 1. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 6 / 22
Welsh s Conjecture A hamilton circuit in a matroid M is a circuit of size r(m) + 1. 9 Conjecture ( Welsh ) If M is a simple regular connected matroid and every cocircuit has at least (r(m) + 1) elements, then M has a hamilton circuit. 1 2 S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 6 / 22
Welsh s Conjecture A hamilton circuit in a matroid M is a circuit of size r(m) + 1. 9 Conjecture ( Welsh ) If M is a simple regular connected matroid and every cocircuit has at least (r(m) + 1) elements, then M has a hamilton circuit. 1 2 Conjecture solved in 1997 by Hochstättler and Jackson. > 30 pages, uses Seymour s decomposition theorem. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 6 / 22
Welsh s Conjecture A hamilton circuit in a matroid M is a circuit of size r(m) + 1. 9 Conjecture ( Welsh ) If M is a simple regular connected matroid and every cocircuit has at least (r(m) + 1) elements, then M has a hamilton circuit. 1 2 Conjecture solved in 1997 by Hochstättler and Jackson. > 30 pages, uses Seymour s decomposition theorem. Shorter proof (9 pages) by McG. in 2010. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 6 / 22
Extending cycles in regular matroids Theorem (McG, 2010) Let M be a simple, connected, regular matroid. Suppose that C is a circuit of M where C 2d 1 for some 2 d r(m)+1 2. C d for all cocircuits C where C C =. Then there is a circuit D such that C D is a larger circuit than C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 7 / 22
Dominating circuits in matroids A circuit C in a matroid M is said to be dominating if each component of M/C has rank at most one. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 8 / 22
Dominating circuits in matroids A circuit C in a matroid M is said to be dominating if each component of M/C has rank at most one. Theorem (McG, 2011) Let M be a simple, connected, regular matroid and let C be a circuit of M where C > r(m) 3 + 1 for all cocircuits C where C C =. Then either C is a dominating circuit or there exists a circuit D for which C D is a larger circuit than C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 8 / 22
Graphic and Cographic cases: an example C 1 * C 2 * C 1 * C 2 * C C S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 9 / 22
C-minimal circuits Given a circuit C in a matroid, a circuit D is said to be C-minimal if C D C, D, and C D are the only circuits in C D. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 10 / 22
C-minimal circuits Given a circuit C in a matroid, a circuit D is said to be C-minimal if C D C, D, and C D are the only circuits in C D. Given a circuit C in matroid, a circuit D is said to be C-augmenting if D is C-mimimal C D > C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 10 / 22
Chord-Circuits Let C be a circuit of a binary matroid M. X (C) is the set of chords of C. e be a chord of C. Let Ce 1 and Ce 2 be the circuits of C {e} which contain e. Ce 1 and Ce 2 are called chord-circuits of C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 11 / 22
Chord Circuits C 1 e and C 2 e C e 2 C e C e 1 S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 12 / 22
Nested Chords For a chord-circuit C i e we let C i i = C i e\{e}. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 13 / 22
Nested Chords For a chord-circuit C i e we let C i i = C i e\{e}. A set X of chords of a circuit C is nested if there exists an ordering x 1, x 2,..., x m of X and j 1, j 2,..., j m {1, 2} for which C j 1 x 1 C j 2 x 2 C jm x m. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 13 / 22
Nested Chords For a chord-circuit C i e we let C i i = C i e\{e}. A set X of chords of a circuit C is nested if there exists an ordering x 1, x 2,..., x m of X and j 1, j 2,..., j m {1, 2} for which C j 1 x 1 C j 2 x 2 C jm x m. C x 1 x 2 x 3 x m S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 13 / 22
C-Cocircuits For a Hamilton circuit C in a matroid M and elements e, f C, we define C ef = E(M)\cl(C\{e, f }), called a C-cocircuit. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 14 / 22
C-Cocircuits For a Hamilton circuit C in a matroid M and elements e, f C, we define C ef = E(M)\cl(C\{e, f }), called a C-cocircuit. e f S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 14 / 22
A key lemma Lemma Suppose M is a connected binary matroid having no F7 -minor and C is a circuit of M. Let D be a C-minimal circuit for which C D is minimum. Then D\C and C D are circuits of M\(C\D) which belong to different components. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 15 / 22
A picture D D\C C D C\D C S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 16 / 22
The minimal counterexample approach We use the minimal counterexample approach to prove: Theorem Let M be a simple, connected, regular matroid. Suppose that C is a circuit of M where C 2d 1 for some 2 d r(m)+1 2. C d for all cocircuits C where C C =. Then there is a circuit D such that C D is a larger circuit than C. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 17 / 22
The minimal counterexample approach We use the minimal counterexample approach to prove: Theorem Let M be a simple, connected, regular matroid. Suppose that C is a circuit of M where C 2d 1 for some 2 d r(m)+1 2. C d for all cocircuits C where C C =. Then there is a circuit D such that C D is a larger circuit than C. A minimum counterexample to the above theorem has the properties: M be a simple, connected, regular matroid. C is a circuit of size less that 2d C d for all cocircuits C where C C =. there is no circuit D such that C D is a larger circuit than C. E(M) is minimum subject to the above. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 17 / 22
Properties of a minimum counterexample There is a C-minimal circuit D for which D\C and C D are circuits of M\(C\D) which belong to different components. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 18 / 22
Properties of a minimum counterexample There is a C-minimal circuit D for which D\C and C D are circuits of M\(C\D) which belong to different components. Let N be the matroid obtained from M by deleting the component in M\(C\D) which contains C D. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 18 / 22
Properties of a minimum counterexample There is a C-minimal circuit D for which D\C and C D are circuits of M\(C\D) which belong to different components. Let N be the matroid obtained from M by deleting the component in M\(C\D) which contains C D. Important property : D can be chosen so that Ω = C D is a Hamilton circuit of N. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 18 / 22
D Ω C S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 19 / 22
Nested chords in Ω Let x be a chord of Ω such that Ω 1 x is minimal. Ω 1 x D. Important property : Assuming M has no R 10 -minor, we can choose e, f Ω 1 x such that Ω ef \{e, f } is a nested set of chords of Ω. S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 20 / 22
e f D x Ω S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 21 / 22
An extension of Welsh s conjecture 21 Conjecture ( McG ) Let M be a simple, connected, regular matroid such that every cocircuit has at least 1 2 (r(m) + 1) elements. Then M has at least αr(m) disjoint Hamilton circuits where α > 0 is a constant not depending on M S. McGuinness (TRU) Hamilton Circuits and Dominating Circuits in Regular Matroids June 2012 22 / 22