Complementary cycles in regular bipartite tournaments: a proof of - PowerPoint PPT Presentation

Definitions and motivations Sketch of the proof Some open questions Complementary cycles in regular bipartite tournaments: a proof of Manoussakis, Song and Zhang conjecture Jocelyn Thiebaut joint work with Stéphane Bessy LIRMM, Montpellier, France JGA 2017, LaBRI, Bordeaux June 7, 2018 [1/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations The structure of cycle factor Definitions cycle factor : partition of the vertices into (directed) vertex-disjoint cycles. t -cycle factor : partition of the vertices into t vertex-disjoint cycles. ( n 1 , n 2 , . . . , n t ) -cycle factor : t -cycle factor whose cycles are of size n 1 , n 2 , . . . , n t . [2/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations The structure of cycle factor Definitions cycle factor : partition of the vertices into (directed) vertex-disjoint cycles. t -cycle factor : partition of the vertices into t vertex-disjoint cycles. ( n 1 , n 2 , . . . , n t ) -cycle factor : t -cycle factor whose cycles are of size n 1 , n 2 , . . . , n t . [2/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations The structure of cycle factor Definitions cycle factor : partition of the vertices into (directed) vertex-disjoint cycles. t -cycle factor : partition of the vertices into t vertex-disjoint cycles. ( n 1 , n 2 , . . . , n t ) -cycle factor : t -cycle factor whose cycles are of size n 1 , n 2 , . . . , n t . [2/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations The classes of digraphs Some more definitions... tournament : orientation of the clique. bipartite tournament : orientation of a complete bipartite graph. k -regular bipartite tournament : bipartite tournament such that ∀ u , d + ( u ) = d − ( u ) = k . (therefore, we have n = 4 k ) [3/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Some results on tournaments [4/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Some results on tournaments Theorem (Camion) 1-cycle factor Let T be a strong tournament. Then T is Hamiltonian (i.e. admits a 1-cycle factor). [4/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Some results on tournaments Theorem (Camion) 1-cycle factor Let T be a strong tournament. Then T is Hamiltonian (i.e. admits a 1-cycle factor). Theorem (Li and Shu) 2-cycle factor Let T be a strong tournament. If max { δ + ( T ) , δ − ( T ) } ≥ 3, | V ( T ) | ≥ 6 and T ≇ P 7 , then T admits a 2-cycle factor. [4/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Some results on tournaments Theorem (Camion) 1-cycle factor Let T be a strong tournament. Then T is Hamiltonian (i.e. admits a 1-cycle factor). Theorem (Li and Shu) 2-cycle factor Let T be a strong tournament. If max { δ + ( T ) , δ − ( T ) } ≥ 3, | V ( T ) | ≥ 6 and T ≇ P 7 , then T admits a 2-cycle factor. Theorem (Reid and Song) ( p , n − p ) -cycle factor Let T be a 2-connected tournament. If | V ( T ) | ≥ 6 and T ≇ P 7 , then for any 3 ≤ p ≤ n − 3, T admits a ( p , n − p ) -cycle factor. [4/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Analogous results on bipartite tournaments Theorem (Camion) 1-cycle factor Let T be a strong tournament. Then T is Hamiltonian (i.e. admits a 1-cycle factor). Theorem (Li and Shu) 2-cycle factor Let T be a strong tournament. If max { δ + ( T ) , δ − ( T ) } ≥ 3, | V ( T ) | ≥ 6 and T ≇ P 7 , then T admits a 2-cycle factor. Theorem (Reid and Song) ( p , n − p ) -cycle factor Let T be a 2-connected tournament. If | V ( T ) | ≥ 6 and T ≇ P 7 , then for any 3 ≤ p ≤ n − 3, T admits a ( p , n − p ) -cycle factor. [5/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Analogous results on bipartite tournaments Theorem (Häggkvist and Manoussakis) 1-cycle factor Let B be a strong bipartite tournament. If B admits a cycle factor then B is Hamiltonian. Theorem (Li and Shu) 2-cycle factor Let T be a strong tournament. If max { δ + ( T ) , δ − ( T ) } ≥ 3, | V ( T ) | ≥ 6 and T ≇ P 7 , then T admits a 2-cycle factor. Theorem (Reid and Song) ( p , n − p ) -cycle factor Let T be a 2-connected tournament. If | V ( T ) | ≥ 6 and T ≇ P 7 , then for any 3 ≤ p ≤ n − 3, T admits a ( p , n − p ) -cycle factor. [5/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Analogous results on bipartite tournaments Theorem (Häggkvist and Manoussakis) 1-cycle factor Let B be a strong bipartite tournament. If B admits a cycle factor then B is Hamiltonian. Theorem (Zhang and Song) 2-cycle factor Let B be a k -regular bipartite tournament. Then B admits a ( 4 , 4 k − 4 ) -cycle factor. Theorem (Reid and Song) ( p , n − p ) -cycle factor Let T be a 2-connected tournament. If | V ( T ) | ≥ 6 and T ≇ P 7 , then for any 3 ≤ p ≤ n − 3, T admits a ( p , n − p ) -cycle factor. [5/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Analogous results on bipartite tournaments Theorem (Häggkvist and Manoussakis) 1-cycle factor Let B be a strong bipartite tournament. If B admits a cycle factor then B is Hamiltonian. Theorem (Zhang and Song) 2-cycle factor Let B be a k -regular bipartite tournament. Then B admits a ( 4 , 4 k − 4 ) -cycle factor. Conjecture (Zhang, Manoussakis and Song) ( p , n − p ) -cycle factor Let B be a k -regular bipartite tournament. If B ≇ F 4 k , then for any 2 ≤ p ≤ 2 k − 2, B admits a ( 2 p , 4 k − 2 p ) -cycle factor. [5/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Cycle factors Sketch of the proof The classes of digraphs Some open questions Motivations Some remarks The forbidden digraph F 4 k : B D A C We have | A | = | B | = | C | = | D | = k . Conjecture known for p = 2 and p = 3. [6/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Some useful tools Sketch of the proof One case of the proof Some open questions One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2 p , then B admits a ( 2 p , 4 k − 2 p ) -cycle factor. Idea of the proof : We merge the cycles of the cycle factor. C 1 C C 1 [7/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Some useful tools Sketch of the proof One case of the proof Some open questions One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2 p , then B admits a ( 2 p , 4 k − 2 p ) -cycle factor. Idea of the proof : We merge the cycles of the cycle factor. C 1 C 2 C C 1 C ℓ [7/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Some useful tools Sketch of the proof One case of the proof Some open questions One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2 p , then B admits a ( 2 p , 4 k − 2 p ) -cycle factor. Idea of the proof : We merge the cycles of the cycle factor. C 1 strong C 2 C C 1 C ℓ [7/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Definitions and motivations Some useful tools Sketch of the proof One case of the proof Some open questions One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2 p , then B admits a ( 2 p , 4 k − 2 p ) -cycle factor. Idea of the proof : We merge the cycles of the cycle factor. C 1 C 2 C C 1 C ℓ [7/12] Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments