Decompositions of Hypergraphs Felix Joos July 2020 Old results - PowerPoint PPT Presentation

Decompositions of Hypergraphs Felix Joos July 2020

Old results Theorem (Walecki, 1892) K 2 n +1 has a decomposition into edge-disjoint Hamilton cycles.

Old results Theorem (Walecki, 1892) K 2 n +1 has a decomposition into edge-disjoint Hamilton cycles. Theorem (Dirac, 1952) Every graph G on n ≥ 3 vertices with δ ( G ) ≥ n 2 contains a Hamilton cycle.

Old results Theorem (Walecki, 1892) K 2 n +1 has a decomposition into edge-disjoint Hamilton cycles. Theorem (Dirac, 1952) Every graph G on n ≥ 3 vertices with δ ( G ) ≥ n 2 contains a Hamilton cycle. Theorem (Hajnal, Szemer´ edi, 1970) k | n and δ ( G ) ≥ k − 1 k n, then G contains a K k -factor.

Generalizations Theorem (Csaba, K¨ uhn, Lo, Osthus, Treglown, 2016) G r-regular with r ≥ n 2 and even, n large, then G has a Hamilton decomposition.

Generalizations Theorem (Csaba, K¨ uhn, Lo, Osthus, Treglown, 2016) G r-regular with r ≥ n 2 and even, n large, then G has a Hamilton decomposition. Theorem (B¨ ottcher, Schacht, Taraz, 2009) χ ( H ) = k, H has bandwidth o ( n ) and ∆( H ) = O (1) δ ( G ) ≥ ( k − 1 + o (1)) n, then H ⊂ G. k

Open problems Conjecture (Nash-Williams, 1970) δ ( G ) ≥ 3 n 4 and G is triangle-divisible, then G has a triangle decomposition

Open problems Conjecture (Nash-Williams, 1970) δ ( G ) ≥ 3 n 4 and G is triangle-divisible, then G has a triangle decomposition Problem Given a graph H, determine δ H where δ H is the least δ such that for every ε > 0 and G on n (large) vertices with δ ( G ) ≥ ( δ + ε ) n has a H-decomposition subject to divisibility conditions?

A step forward Fractional H -decomposition of G : ω : { copies of H in G } → [0 , 1] such that � H ∋ e ω ( H ) = 1 for e ∈ E ( G ).

A step forward Fractional H -decomposition of G : ω : { copies of H in G } → [0 , 1] such that � H ∋ e ω ( H ) = 1 for e ∈ E ( G ). δ ∗ H fractional version of δ H .

A step forward Fractional H -decomposition of G : ω : { copies of H in G } → [0 , 1] such that � H ∋ e ω ( H ) = 1 for e ∈ E ( G ). δ ∗ H fractional version of δ H . Theorem (Barber, K¨ uhn, Lo, Osthus, 2016; Glock, K¨ uhn, Lo, Montgomery, Osthus, 2019) H , 1 − 1 1 δ H ∈ { δ ∗ χ , 1 − χ +1 } for χ = χ ( H ) ≥ 5 solved for bipartite H General tool: turning fractional decompositions into decompositions

Cycles Theorem (Barber, K¨ uhn, Lo, Osthus, 2016) δ C 4 = 2 3 and δ C 2 ℓ = 1 2 for ℓ ≥ 3 δ C 2 ℓ +1 = δ ∗ C 2 ℓ +1

Cycles Theorem (Barber, K¨ uhn, Lo, Osthus, 2016) δ C 4 = 2 3 and δ C 2 ℓ = 1 2 for ℓ ≥ 3 δ C 2 ℓ +1 = δ ∗ C 2 ℓ +1 uhn, 2020 + ) Theorem (J., M. K¨ δ C 2 ℓ +1 → 1 2 ( ℓ → ∞ )

Hypergraphs - old results G k -uniform ( k -graph): edges of size k d m ( S ) = number of edges containing S for | S | = m δ m ( G ) = min S d m ( S ) δ ( G ) = δ k − 1 ( G )

Hypergraphs - old results G k -uniform ( k -graph): edges of size k d m ( S ) = number of edges containing S for | S | = m δ m ( G ) = min S d m ( S ) δ ( G ) = δ k − 1 ( G ) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge

Hypergraphs - old results G k -uniform ( k -graph): edges of size k d m ( S ) = number of edges containing S for | S | = m δ m ( G ) = min S d m ( S ) δ ( G ) = δ k − 1 ( G ) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) δ ( G ) ≥ ( 1 2 + o (1)) n, then G contains a (tight) Hamilton cycle

Hypergraphs - old results G k -uniform ( k -graph): edges of size k d m ( S ) = number of edges containing S for | S | = m δ m ( G ) = min S d m ( S ) δ ( G ) = δ k − 1 ( G ) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) δ ( G ) ≥ ( 1 2 + o (1)) n, then G contains a (tight) Hamilton cycle Theorem (Lang, Sahueza-Matamala; Polcyn, Reiher, R¨ odl, ulke, 2020 + ) Sch¨ δ k − 2 ( G ) ≥ ( 5 9 + o (1)) n 2 / 2 , then G contains a (tight) Hamilton cycle

New results

New results Theorem (J., K¨ uhn, 2020 + ) δ ∗ → 1 2 for ℓ → ∞ C ( k ) ℓ

New results Theorem (J., K¨ uhn, 2020 + ) δ ∗ → 1 2 for ℓ → ∞ C ( k ) ℓ Proof method: Restriction systems + random walks

Proof Sketch non - bipahte , counected graph 1- Simple random walk - la reg on a , ⇒ Uniform limit distribution line graph a d- regulaer the Z Find subgraph in > & " bes der dd " " " : reshiction system ( avoid a rete ) Markov chain ordere d around edges walking 3- on " from ötof =p d to Fish steps ] =p # b- walks P [ traue ⇐ > 4- h h with the conform restriktion system in b) pm → 1- 5 ( very quichly Zeh ) - ↳ x Kind e and f ! of .

New results II

New results II Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) ∀ ǫ > 0 , k ∈ N the following holds for all large n: G k-graph with δ ( G ) ≥ ( 1 2 + ε ) n, then G contains a (tight) Hamilton cycle

New results II Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) ∀ ǫ > 0 , k ∈ N the following holds for all large n: G k-graph with δ ( G ) ≥ ( 1 2 + ε ) n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n:

New results II Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) ∀ ǫ > 0 , k ∈ N the following holds for all large n: G k-graph with δ ( G ) ≥ ( 1 2 + ε ) n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n

New results II Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) ∀ ǫ > 0 , k ∈ N the following holds for all large n: G k-graph with δ ( G ) ≥ ( 1 2 + ε ) n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n • | d 1 ( v ) − d 1 ( u ) | ≤ ǫ ′ n k − 1

New results II Theorem (R¨ odl, Ruci´ nski, Szemer´ edi, 2008) ∀ ǫ > 0 , k ∈ N the following holds for all large n: G k-graph with δ ( G ) ≥ ( 1 2 + ε ) n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n • | d 1 ( v ) − d 1 ( u ) | ≤ ǫ ′ n k − 1 then G contains ( e ( G ) − ǫ n k ) / n edge-disjoint Hamilton cycles.

New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n • | d 1 ( v ) − d 1 ( u ) | ≤ ǫ ′ n k − 1 then G contains ( e ( G ) − ǫ n k ) / n edge-disjoint Hamilton cycles.

New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n • | d 1 ( v ) − d 1 ( u ) | ≤ ǫ ′ n k − 1 then G contains ( e ( G ) − ǫ n k ) / n edge-disjoint Hamilton cycles. Corollary Vertex-regular k-graphs G with δ ( G ) ≥ ( 1 2 + o (1)) n can be approximately decomposed into Hamilton cycles with an arbitrary good precision.

New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020 + ) ∀ ǫ > 0 , k ∈ N ∃ ǫ ′ > 0 such that for all large n: • G k-graph with δ ( G ) ≥ ( 1 2 + ǫ ) n • | d 1 ( v ) − d 1 ( u ) | ≤ ǫ ′ n k − 1 then G contains ( e ( G ) − ǫ n k ) / n edge-disjoint Hamilton cycles. Corollary Vertex-regular k-graphs G with δ ( G ) ≥ ( 1 2 + o (1)) n can be approximately decomposed into Hamilton cycles with an arbitrary good precision. Proof method: δ ∗ C ℓ ≤ 1 2 + ǫ for large enough ℓ ; random process; absorption

Summary I

Summary I • Fractional ℓ -cycle decompositions

Summary I • Fractional ℓ -cycle decompositions • approx. decompositions into Hamilton cycles

Summary I • Fractional ℓ -cycle decompositions • approx. decompositions into Hamilton cycles in hypergraphs under very weak assumptions

Graph decompositions Three conjectures: ◮ Ringel: K 2 n +1 into any tree with n edges ◮ Tree packing conj.: K n into trees T 1 ,..., T n − 1 with e ( T i ) = i ◮ Oberwolfach problem: K 2 n +1 into any spanning union of cycles

Graph decompositions - progress Approximate decompositions: • ∆ = O (1), trees, almost spanning, K n B¨ ottcher, Hladk´ y, Piguet, Taraz, 16

Graph decompositions - progress Approximate decompositions: • ∆ = O (1), trees, almost spanning, K n B¨ ottcher, Hladk´ y, Piguet, Taraz, 16 • ∆ = O (1), separable, almost spanning, K n Messuti, R¨ odl, Schacht, 17

Graph decompositions - progress Approximate decompositions: • ∆ = O (1), trees, almost spanning, K n B¨ ottcher, Hladk´ y, Piguet, Taraz, 16 • ∆ = O (1), separable, almost spanning, K n Messuti, R¨ odl, Schacht, 17 • ∆ = O (1), separable, spanning, K n Ferber, Lee, Mousset, 16