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

[1/12] 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

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[2/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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. (n1, n2, . . . , nt)-cycle factor : t-cycle factor whose cycles are of size n1, n2, . . . , nt.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[2/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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. (n1, n2, . . . , nt)-cycle factor : t-cycle factor whose cycles are of size n1, n2, . . . , nt.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[2/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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. (n1, n2, . . . , nt)-cycle factor : t-cycle factor whose cycles are of size n1, n2, . . . , nt.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[3/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs Motivations

The classes of digraphs Some more definitions... tournament :

• rientation of the clique.

bipartite tournament :

• rientation of a

complete bipartite graph. k-regular bipartite tournament : bipartite tournament such that ∀u, d+(u) = d−(u) = k. (therefore, we have n = 4k)

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[4/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs Motivations

Some results on tournaments

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[4/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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).

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[4/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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 ≇ P7, then T admits a 2-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[4/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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 ≇ P7, 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 ≇ P7, then for any 3 ≤ p ≤ n − 3, T admits a (p, n − p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[5/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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 ≇ P7, 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 ≇ P7, then for any 3 ≤ p ≤ n − 3, T admits a (p, n − p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[5/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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 ≇ P7, 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 ≇ P7, then for any 3 ≤ p ≤ n − 3, T admits a (p, n − p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[5/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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, 4k − 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 ≇ P7, then for any 3 ≤ p ≤ n − 3, T admits a (p, n − p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[5/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs 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, 4k − 4)-cycle factor.

Conjecture (Zhang, Manoussakis and Song)

(p, n − p)-cycle factor Let B be a k-regular bipartite tournament. If B ≇ F4k, then for any 2 ≤ p ≤ 2k − 2, B admits a (2p, 4k − 2p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[6/12] Definitions and motivations Sketch of the proof Some open questions Cycle factors The classes of digraphs Motivations

Some remarks The forbidden digraph F4k :

A B C D

We have |A| = |B| = |C| = |D| = k. Conjecture known for p = 2 and p = 3.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C strong C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C z1 y1yℓ zℓ b1 r1 b2 r2 C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C z1 y1yℓ zℓ b1 r1 b2 r2 C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C z1 y1yℓ zℓ b1 r1 b2 r2 C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[7/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

One Claim: From a cycle factor to a 2-cfycle factor Claim If B has a cycle factor containing a cycle of length 2p, then B admits a (2p, 4k − 2p)-cycle factor. Idea of the proof : We merge the cycles of the cycle factor.

C1 C2 Cℓ C z1 y1yℓ zℓ b1 r1 b2 r2 C1

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s1 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Properties: 2k vertices; k-regular;

[8/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The contracted digraph according to a perfect matching

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Properties: 2k vertices; k-regular; cycle factor.

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

∀x ∈ C, N+

BM′(x) = N+ BM(C −(x))

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM ′ S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

∀x ∈ C, N+

BM′(x) = N+ BM(C −(x))

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM ′ S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM ′ S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[9/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

The anti-cycles in the contracted digraph

B BM ′ S T

s0 t0 s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s0 s1 s2 s3 s5 s4 Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

The switch modified the chosen matching and so the structure of the cycle factor.

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C

(p vertices)

C′

(2k − p vertices)

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM′

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM′ Häggkvist and Manoussakis’s theorem on red vertices + previous claim

[10/12] Definitions and motivations Sketch of the proof Some open questions Some useful tools One case of the proof

Case when we have two anti-paths

u′ v′ u v

C C′

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

Induction on p: (p, 2k − p)-cycle factor (p + 1, 2k − p − 1)-cycle factor.

BM Remaining case: we do not have such anti-paths...

[11/12] Definitions and motivations Sketch of the proof Some open questions

What’s next? Theorem (Bessy and T.) Let B be a k-regular bipartite tournament. If B ≇ F4k, then for any 2 ≤ p ≤ 2k − 2, B admits a (2p, 4k − 2p)-cycle factor.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[11/12] Definitions and motivations Sketch of the proof Some open questions

What’s next? Theorem (Bessy and T.) Let B be a k-regular bipartite tournament. If B ≇ F4k, then for any 2 ≤ p ≤ 2k − 2, B admits a (2p, 4k − 2p)-cycle factor. Does B admits the same property if we replace the regularity by cycle-factor + k-vertex connectivity?

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[11/12] Definitions and motivations Sketch of the proof Some open questions

What’s next? Theorem (Bessy and T.) Let B be a k-regular bipartite tournament. If B ≇ F4k, then for any 2 ≤ p ≤ 2k − 2, B admits a (2p, 4k − 2p)-cycle factor. Does B admits the same property if we replace the regularity by cycle-factor + k-vertex connectivity? Under which conditions does B admit a (p1, p2, . . . , pℓ)-cycle factor? Theorem (Kühn, Ostus and Townsend) If T is a O(ℓ4logℓ)-connected tournament then for every p1 + · · · + pℓ = n, (pi ≥ 3) T has a (p1, p2, . . . , pℓ)-cycle factor

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[11/12] Definitions and motivations Sketch of the proof Some open questions

What’s next? Theorem (Bessy and T.) Let B be a k-regular bipartite tournament. If B ≇ F4k, then for any 2 ≤ p ≤ 2k − 2, B admits a (2p, 4k − 2p)-cycle factor. Does B admits the same property if we replace the regularity by cycle-factor + k-vertex connectivity? Under which conditions does B admit a (p1, p2, . . . , pℓ)-cycle factor? What happens B is a 3-partite, 4-partite, in short, c-partite digraph?

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[12/12] Definitions and motivations Sketch of the proof Some open questions

Thank you for your attention.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments

[13/12] Definitions and motivations Sketch of the proof Some open questions

Thank you for your attention.

Jocelyn Thiebaut Complementary cycles in regular bipartite tournaments