Eulerian hypergraphs 31st Cumberland Conference@ UCF Songling Shan - PowerPoint PPT Presentation

Eulerian hypergraphs 31st Cumberland Conference@ UCF Songling Shan May 18, 2019 Illinois State University Joint work with Amin Bahmanian

Preliminary

Hypergraph, minimum ℓ -degree, and more Hypergraph: H = ( V , E ), e ∈ E , e ⊆ V H is k -uniform, if | e | = k , ∀ e ∈ E d H ( S ): number of edges containing S , write d H ( v ) if S = { v } k -uniform H , 1 ≤ ℓ ≤ k δ ℓ ( H ): smallest d H ( S ) for all H is 3-uniform S ⊆ V , | S | = ℓ δ 1 ( H ) = 1 δ 1 ( H ): minimum vertex degree d H ( { u , v } ) = 0 δ k − 1 ( H ): minimum codegree δ 2 ( H ) = 0 1

Euler tour in H A trail in H : an alternating sequence v 0 e 1 v 1 e 2 v 2 · · · v m − 1 e m v m of vertices and edges in H such that e i � = e j , i � = j v i − 1 , v i ∈ e i , v i − 1 � = v i An Euler trail in H : a trail containing all edges of H An Euler tour in H : a closed trail ( v 0 = v m ) containing all edges of H H is Eulerian: if H has an Euler tour An Euler tour does not need to span V ! 2

Strong cutedge in H Strong cutedge in H : e ∈ E such that c ( H − e ) = c ( H ) + | e | − 1 Analogous to cutedge in graphs If H has a strong cutedge, then H has no Euler tour 3

Existence of Euler tour in 3-uniform hypergraph

Necessary conditions Theorem (Lonc and Naroski, 2010) If H = ( V , E ) is a k-uniform hypergraph that admits an Euler tour, then 1 H has no strong cutedge, 2 | E | ≤ � ⌊ d H ( v ) / 2 ⌋ , v ∈ V 3 | V odd | ≤ � ( | e | − 2) = e ∈ E ( k − 2) | E | . 4

Euler tour in 3-uniform hypergraph Theorem (Lonc and Naroski, 2010) Let H = ( V , E ) be a k-uniform hypergraph with a connected ( k − 1) - intersection graph. Then H is Eulerian if and only if | V odd | ≤ ( k − 2) | E | . 5

NP-completeness of determination Theorem (Lonc and Naroski, 2010) Let k > 2 . The problem of determining if a given k-uniform hypergraph has an Euler tour is NP-complete. 6

NP-completeness of determination Theorem (Lonc and Naroski, 2010) Let k > 2 . The problem of determining if a given k-uniform hypergraph has an Euler tour is NP-complete. ⇐ ⇒ Euler tour problem in Hamiltonian cycle problem in cubic graph k -uniform hypergraph NP-complete; Garey, Johnson, and Tarjan, 1976 6

NP-completeness of determination G 4-uniform H 7

NP-completeness of determination G 4-uniform H Hamiltonian cycle: xuvwx 7

NP-completeness of determination G 4-uniform H xue 1 uve 2 vwe 3 wxe 4 xu Hamiltonian cycle: xuvwx 7

Euler tour in 3-uniform hypergraphs Theorem (ˇ Sajna, Wagner, 2016) 1 Every Steiner triple system with at least two triples is Eulerian. 2 Every 3-uniform hypergraph H with δ 2 ( H ) ≥ 2 is Eulerian. 8

Euler tour in 3-uniform hypergraphs Theorem (ˇ Sajna, Wagner, 2016) 1 Every Steiner triple system with at least two triples is Eulerian. 2 Every 3-uniform hypergraph H with δ 2 ( H ) ≥ 2 is Eulerian. Inductive Lemma: Let h , k be positive integers with k ≥ h ≥ 3. If any h -uniform hypergraph H 1 with δ t ( H 1 ) ≥ r is Eulerian, then any k -uniform hypergraph H 2 with δ k − h + t ( H 2 ) ≥ r is Eulerian. 8

Euler tour in 3-uniform hypergraphs Theorem (ˇ Sajna, Wagner, 2016) 1 Every Steiner triple system with at least two triples is Eulerian. 2 Every 3-uniform hypergraph H with δ 2 ( H ) ≥ 2 is Eulerian. Inductive Lemma: Let h , k be positive integers with k ≥ h ≥ 3. If any h -uniform hypergraph H 1 with δ t ( H 1 ) ≥ r is Eulerian, then any k -uniform hypergraph H 2 with δ k − h + t ( H 2 ) ≥ r is Eulerian. By (2) and Inductive Lemma: 3-uniform H with δ 2 ( H ) ≥ 2 is Eulerian ⇒ 4-uniform H with δ 3 ( H ) ≥ 2 is Eulerian ⇒ 5-uniform H with δ 4 ( H ) ≥ 2 is Eulerian · · · ⇒ k -uniform H with δ k − 1 ( H ) ≥ 2 is Eulerian 8

Existence of Euler tour in general hypergraph

Euler tour in general hypergraph Theorem (Bahmanian, S., 2019+) Let k ≥ 4 and H = ( V , E ) be a hypergraph with | V | = n, rk ( H ) = k, and δ 2 ( H ) ≥ k. Then for any distinct vertices u , v ∈ V , H has a spanning Euler trail starting at u and ending at v if H satisfies one of the following conditions. 1 H is k-uniform and n ≥ k 2 2 + k 2 ; 2 H has no multiple edge, cr ( H ) ≥ 3 , and n ≥ k 2 k + k 2 2 + k 2 . Corollary: Let k ≥ 4 and H = ( V , E ) be a k -uniform hypergraph with | V | = n ≥ k 2 2 + k 2 . If δ k − 2 ( H ) ≥ 4, then H admits an Euler tour. 9

Outline of proof Theorem (Bahmanian, S., 2019+) Let k ≥ 4 and H = ( V , E ) be a hypergraph with | V | = n, rk ( H ) = k, and δ 2 ( H ) ≥ k. Then for any distinct vertices u , v ∈ V , H has a spanning Euler trail starting at u and ending at v if H satisfies one of the following conditions. 1 H is k-uniform and n ≥ k 2 2 + k 2 ; 2 H has no multiple edge, cr ( H ) ≥ 3 , and n ≥ k 2 k + k 2 2 + k 2 . Turn the problem of finding a spanning trail in H into a problem of finding a special subgraph in a graph G associated with H 10

Incidence graph of H 11

Incidence graph of H 11

Incidence graph of H Let G [ X , Y ] be bipartite. An X -saturating factor is an even subgraph H of G with d H ( x ) = 2 for any x ∈ X . 11

Incidence graph of H Finding an Euler tour in hypergraph H ⇐ ⇒ I ( H )[ E , V ] has a connected E -saturating factor 11

X -saturating factor in G [ X , Y ] Let G be a graph, f , g : V ( G ) → N with g ( v ) ≤ f ( v ) for any v ∈ V ( G ). A ( g , f )-factor in G is a subgraph H such that for any v ∈ V ( G ), g ( v ) ≤ d H ( v ) ≤ f ( v ). 12

X -saturating factor in G [ X , Y ] Theorem (Parity ( g , f ) -factor Theorem; Lov´ asz, 1972) Let G be a graph and let f , g : V ( G ) → N be functions such that g ( x ) ≤ f ( x ) and g ( x ) ≡ f ( x ) ( mod 2) for all x ∈ V ( G ) . Then G has a ( g , f ) -factor F such that d F ( x ) ≡ f ( x ) ( mod 2) for all x ∈ V ( G ) if and only if for any S , T ⊆ V ( G ) with S ∩ T = ∅ , � � ( d G ( x ) − g ( x )) − e G ( S , T ) − q ( S , T ) ≥ 0 , f ( x ) + x ∈ S x ∈ T where q ( S , T ) is the number of components D of G − ( S ∪ T ) such that � f ( x ) + e G ( V ( D ) , T ) x ∈ V ( D ) is odd. 12

X -saturating factor in G [ X , Y ] 12

X -saturating factor in G [ X , Y ] Corollary ( E -Saturating factor Lemma) Let H = ( V , E ) be a hypergraph and G = I ( H )[ E , V ] be its incidence graph. Then G has an E-saturating factor if and only if for any disjoint S ⊆ E and T ⊆ E ∪ V , � 2 | S | − 2 | T ∩ E | + d G − S ( x ) − q ( S , T ) ≥ 0 , x ∈ T where q ( S , T ) is the number of components D of G − ( S ∪ T ) such that e G ( V ( D ) , T ) is odd. 12

Connected E -saturating factor in I ( H ) Step 1: Take a spanning tree T in I ( H )[ E , V ] such that d T ( e ) ≤ 2, for any e ∈ E . 13

Connected E -saturating factor in I ( H ) Step 2: Obtain T ∗ by deleting from T the leaves that are contained in E T ∗ is still spanning on V 13

Connected E -saturating factor in I ( H ) Step 3: Obtain I ( H ) ∗ by deleting from T ∗ degree 2 vertices that are contained in E and adding some new E -vertices that span on odd degree vertices in T ∗ 13

Connected E -saturating factor in I ( H ) Step 4: Find a E ∗ -saturating factor F ∗ in I ( H ) ∗ 13

Connected E -saturating factor in I ( H ) Step 5: T ∗ ∪ ( F ∗ − { w 1 , w 2 } ) gives a connected E -saturating factor in I ( H ) 13

Quasi-eulerian hypergraph

Quasi-eulerian 3-uniform hypergraph A hypergraph H = ( V , E ) is quasi-eulerian if E can be decomposed into edge-disjoint closed trails in G . Theorem (Bahmanian, ˇ Sajna, 2017 ) Let H = ( V , E ) be a 3-uniform hypergraph without cut edges. Then H is quasi-eulerian. 14

Quasi-eulerian hypergraph Theorem (Bahmanian, S., 2019+ ) Let H = ( V , E ) be a hypergraph such that for every e ∈ E, 3 ≤ c ≤ | e | ≤ d, where c and d are some integers. Then H is quasi-eulerian if H is at least (1 + ⌈ d / c ⌉ ) -edge-connected. 15

Quasi-eulerian hypergraph Theorem (Bahmanian, S., 2019+ ) Let H = ( V , E ) be a hypergraph such that for every e ∈ E, 3 ≤ c ≤ | e | ≤ d, where c and d are some integers. Then H is quasi-eulerian if H is at least (1 + ⌈ d / c ⌉ ) -edge-connected. Corollary: Let k ≥ 3. Every k -uniform hypergraph without cutedge is quasi-eulerian. 15

Open problems

Open problems 1 Let H be a 4-uniform hypergraph with δ 2 ( H ) ≥ 2. Is H Eulerian? 2 Let k ≥ 3. Is there a constant 0 < c < 1 such that every k -uniform hypergraph with δ 1 ( H ) ≥ cn is Eulerian? 3 (Conjecture, Glock, Joos, K˝ uhn, Osthus, 2008) For all k > 2 and ǫ > 0, there exists n 0 ∈ N such that every k -uniform hypergraph H on n ≥ n 0 vertices with δ k − 1 ( H ) ≥ ( 1 2 + ǫ ) n has a tight Euler tour if all vertex degrees are divisible by k . 16

Thank You Thank you for your attention! 17