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 dH(S): number of edges containing S, write dH(v) if S = {v} k-uniform H, 1 ≤ ℓ ≤ k δℓ(H): smallest dH(S) for all S ⊆ V , |S| = ℓ δ1(H): minimum vertex degree δk−1(H): minimum codegree H is 3-uniform δ1(H) = 1 dH({u, v}) = 0 δ2(H) = 0

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Euler tour in H

A trail in H: an alternating sequence v0e1v1e2v2 · · · vm−1emvm of vertices and edges in H such that ei = ej, i = j vi−1, vi ∈ ei, vi−1 = vi An Euler trail in H: a trail containing all edges of H An Euler tour in H: a closed trail (v0 = vm) containing all edges of H H is Eulerian: if H has an Euler tour An Euler tour does not need to span V !

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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

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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| ≤ v∈V

⌊dH(v)/2⌋,

3 |Vodd| ≤ e∈E

(|e| − 2) = (k − 2)|E|.

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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 |Vodd| ≤ (k − 2)|E|.

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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.

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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. Hamiltonian cycle problem in cubic graph ⇐ ⇒ Euler tour problem in k-uniform hypergraph NP-complete; Garey, Johnson, and Tarjan, 1976

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NP-completeness of determination

G 4-uniform H

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NP-completeness of determination

G 4-uniform H Hamiltonian cycle: xuvwx

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NP-completeness of determination

G 4-uniform H Hamiltonian cycle: xuvwx xue1uve2vwe3wxe4xu

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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 δ2(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 δ2(H) ≥ 2 is Eulerian.

Inductive Lemma: Let h, k be positive integers with k ≥ h ≥ 3. If any h-uniform hypergraph H1 with δt(H1) ≥ r is Eulerian, then any k-uniform hypergraph H2 with δk−h+t(H2) ≥ r is Eulerian.

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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 δ2(H) ≥ 2 is Eulerian.

Inductive Lemma: Let h, k be positive integers with k ≥ h ≥ 3. If any h-uniform hypergraph H1 with δt(H1) ≥ r is Eulerian, then any k-uniform hypergraph H2 with δk−h+t(H2) ≥ r is Eulerian. By (2) and Inductive Lemma: 3-uniform H with δ2(H) ≥ 2 is Eulerian⇒ 4-uniform H with δ3(H) ≥ 2 is Eulerian ⇒ 5-uniform H with δ4(H) ≥ 2 is Eulerian · · · ⇒ k-uniform H with δk−1(H) ≥ 2 is Eulerian

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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 δ2(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 ≥ k2 2 + k 2; 2 H has no multiple edge, cr(H) ≥ 3, and n ≥ k2k + k2 2 + k 2.

Corollary: Let k ≥ 4 and H = (V , E) be a k-uniform hypergraph with |V | = n ≥ k2

2 + k

• 2. If δk−2(H) ≥ 4, then H admits an Euler tour.

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Outline of proof

Theorem (Bahmanian, S., 2019+) Let k ≥ 4 and H = (V , E) be a hypergraph with |V | = n, rk(H) = k, and δ2(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 ≥ k2 2 + k 2; 2 H has no multiple edge, cr(H) ≥ 3, and n ≥ k2k + k2 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

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Incidence graph of H

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Incidence graph of H

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Incidence graph of H

Let G[X, Y ] be bipartite. An X-saturating factor is an even subgraph H

• f G with dH(x) = 2 for any x ∈ X.

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Incidence graph of H

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

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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) ≤ dH(v) ≤ f (v).

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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 dF(x) ≡ f (x) (mod 2) for all x ∈ V (G) if and

• nly if for any S, T ⊆ V (G) with S ∩ T = ∅,
• x∈S

f (x) +

• x∈T

(dG(x) − g(x)) − eG(S, T) − q(S, T) ≥ 0, where q(S, T) is the number of components D of G − (S ∪ T) such that

• x∈V (D)

f (x) + eG(V (D), T) is odd.

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X-saturating factor in G[X, Y ]

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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| +

• x∈T

dG−S(x) − q(S, T) ≥ 0, where q(S, T) is the number of components D of G − (S ∪ T) such that eG(V (D), T) is odd.

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Connected E-saturating factor in I(H)

Step 1: Take a spanning tree T in I(H)[E, V ] such that dT(e) ≤ 2, for any e ∈ E.

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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

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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 ∗

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Connected E-saturating factor in I(H)

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

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Connected E-saturating factor in I(H)

Step 5: T ∗ ∪ (F ∗ − {w1, w2}) gives a connected E-saturating factor in I(H)

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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.

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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.

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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.

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Open problems

Open problems

1 Let H be a 4-uniform hypergraph with δ2(H) ≥ 2. Is H Eulerian? 2 Let k ≥ 3. Is there a constant 0 < c < 1 such that every k-uniform

hypergraph with δ1(H) ≥ cn is Eulerian?

3 (Conjecture, Glock, Joos, K˝

uhn, Osthus, 2008) For all k > 2 and ǫ > 0, there exists n0 ∈ N such that every k-uniform hypergraph H

• n n ≥ n0 vertices with δk−1(H) ≥ ( 1

2 + ǫ)n has a tight Euler tour if

all vertex degrees are divisible by k.

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Thank You

Thank you for your attention!

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