SLIDE 1 Finding many edge-disjoint Hamiltonian cycles in dense graphs
Stephen G. Hartke
Department of Mathematics University of Nebraska–Lincoln www.math.unl.edu/∼shartke2 hartke@math.unl.edu Joint work with T yler Seacrest
SLIDE 2 Finding Edge-Disj. Subgraphs in Dense Graphs
- Ques. How many edge-disjoint 1-factors can we find
in a dense graph G with minimum degree δ ≥ n/2? How many edge-disjoint Hamiltonian cyles?
SLIDE 3 Finding Edge-Disj. Subgraphs in Dense Graphs
- Ques. How many edge-disjoint 1-factors can we find
in a dense graph G with minimum degree δ ≥ n/2? How many edge-disjoint Hamiltonian cyles?
- Thm. [Katerinis 1985; Egawa and Enomoto 1989]
If δ ≥ n/2, then G contains a k-factor for all k ≤ (n + 5)/4. This is best possible. Thus, roughly n/4 edge-disjoint 1-factors and n/8 edge-disjoint Ham cycles are the best we can hope for.
SLIDE 4 Finding Edge-Disj. Subgraphs in Dense Graphs
- Ques. How many edge-disjoint 1-factors can we find
in a dense graph G with minimum degree δ ≥ n/2? How many edge-disjoint Hamiltonian cyles?
- Thm. [Katerinis 1985; Egawa and Enomoto 1989]
If δ ≥ n/2, then G contains a k-factor for all k ≤ (n + 5)/4. This is best possible. Thus, roughly n/4 edge-disjoint 1-factors and n/8 edge-disjoint Ham cycles are the best we can hope for.
- Thm. [Nash-Williams, 1971]
If δ ≥ n/2, then G contains at least ⌊5n/224⌋ edge-disjoint Hamiltonian cycles. This gives roughly n/46 edge-disjoint Hamiltonian cycles, which gives n/23 edge-disjoint 1-factors.
SLIDE 5 Results
- Thm. [Christofides, Kühn, Osthus, 2011+]
For every ε > 0, if n is sufficiently large and δ ≥ (1/2 + ε)n, then G contains n/8 Hamiltonian cycles.
SLIDE 6 Results
- Thm. [Christofides, Kühn, Osthus, 2011+]
For every ε > 0, if n is sufficiently large and δ ≥ (1/2 + ε)n, then G contains n/8 Hamiltonian cycles. Under the hypotheses, this gives n/4 edge-disjoint 1-factors. CKO’s proof uses the Regularity Lemma.
SLIDE 7 Results
- Thm. [Christofides, Kühn, Osthus, 2011+]
For every ε > 0, if n is sufficiently large and δ ≥ (1/2 + ε)n, then G contains n/8 Hamiltonian cycles. Under the hypotheses, this gives n/4 edge-disjoint 1-factors. CKO’s proof uses the Regularity Lemma.
Let G have min deg δ ≥ n/2 + O(n3/4 ln(n)). Then G contains n/8 − O(n7/8 ln(n)) edge-disjoint Ham cycles. Our proof avoids the Regularity Lemma and hence the constants are much smaller.
SLIDE 8 Results
- Thm. [Christofides, Kühn, Osthus, 2011+]
For every ε > 0, if n is sufficiently large and δ ≥ (1/2 + ε)n, then G contains n/8 Hamiltonian cycles. Under the hypotheses, this gives n/4 edge-disjoint 1-factors. CKO’s proof uses the Regularity Lemma.
Let G have min deg δ ≥ n/2 + O(n3/4 ln(n)). Then G contains n/8 − O(n7/8 ln(n)) edge-disjoint Ham cycles. Our proof avoids the Regularity Lemma and hence the constants are much smaller. (All of our theorems also describe when δ > n/2.)
SLIDE 9 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
with at least half the edges of G.
SLIDE 10 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
with at least half the edges of G. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part.
SLIDE 11 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
with at least half the edges of G. Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part. Repeat; the number of edges in H strictly increases.
SLIDE 12 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
where degH() ≥ degG()/2 for all vertices . Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part. Repeat; the number of edges in H strictly increases.
SLIDE 13 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
where degH() ≥ degG()/2 for all vertices . Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part. Repeat; the number of edges in H strictly increases.
- Prop. Every graph G has a bipartite subgraph H
with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[|E(H)|] ≥ |E(G)|/2.
SLIDE 14 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
where degH() ≥ degG()/2 for all vertices . Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part. Repeat; the number of edges in H strictly increases.
- Prop. Every graph G has a bipartite subgraph H
with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[|E(H)|] ≥ |E(G)|/2.
- The same analysis works with a random balanced partition.
SLIDE 15 Large Bipartite Subgraphs
- Prop. Every graph G has a bipartite subgraph H
where degH() ≥ degG()/2 for all vertices . Proof. Partition verts of G; let H be the induced bipartite subgraph. If has degH() < degG()/2, then push to the other part. Repeat; the number of edges in H strictly increases.
- Prop. Every graph G has a balanced bipartite subgraph H
with at least half the edges of G. Proof 2 (sketch). Randomly partition the vertices of G to form H. Then E[|E(H)|] ≥ |E(G)|/2.
- The same analysis works with a random balanced partition.
SLIDE 16
Large Bipartite Subgraphs
Can both properties be simultaneously obtained?
SLIDE 17 Large Bipartite Subgraphs
Can both properties be simultaneously obtained? A bisection is a balanced spanning bipartite subgraph.
- Conj. [Bollobás–Scott 2002]
Every graph G with an even number of vertices has a bisection H with degH() ≥ ⌊degG()/2⌋ for all .
SLIDE 18
Sharpness
The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with degH() ≥ ⌊degG()/2⌋ + 1 for all . Let k < n/2. Kk n − k Kk ∨ (n − k)K1
SLIDE 19
Sharpness
The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with degH() ≥ ⌊degG()/2⌋ + 1 for all . Let k < n/2. Kk n − k Kk ∨ (n − k)K1 For any partition, both partite sets have vertices from the independent set.
SLIDE 20 Probabilistic Approach
- Thm. Any graph G on n (even) vertices
has a bisection H such that for all vertices , deg H() ≥ 1 2 deg G() −
[similar result by A. Bush 2009]
SLIDE 21 Probabilistic Approach
- Thm. Any graph G on n (even) vertices
has a bisection H such that for all vertices , deg H() ≥ 1 2 deg G() −
[similar result by A. Bush 2009] Proof Outline. Arbitrarily pair the vertices. Randomly split each pair between the two partite sets.
SLIDE 22 Probabilistic Approach
- Thm. Any graph G on n (even) vertices
has a bisection H such that for all vertices , deg H() ≥ 1 2 deg G() −
[similar result by A. Bush 2009] Proof Outline. Arbitrarily pair the vertices. Randomly split each pair between the two partite sets. Bound the probability of a bad vertex using Chernoff bounds. Combine using the union sum bound.
SLIDE 23 Larger Partitions
- Thm. Let G be a graph on n vertices, where n = pq for p > 1.
Then there exists a partition of G into q parts of size p such that every vertex has at least deg()/q −
- deg() · ln(n) neighbors in each part.
SLIDE 24 Larger Partitions
- Thm. Let G be a graph on n vertices, where n = pq for p > 1.
Then there exists a partition of G into q parts of size p such that every vertex has at least deg()/q −
- deg() · ln(n) neighbors in each part.
An upper bound can be obtained following a construction of Doerr and Srivastav 2003 from discrepancy theory. It is based on a construction of Spencer 1985 using Hadamard matrices.
- Thm. For infinitely many n, there exists a graph G on n vertices
such than any partition of G into q parts contains a part P and vertex such that has less than deg()/q − 1
3
SLIDE 25 Finding k-Factors in Bipartite Graphs
Let G be balanced bipart. graph on 2p vertices with min deg δ ≥ p
2.
Let α = δ +
2 . Then G has an ⌊α⌋-regular spanning subgraph.
SLIDE 26 Finding k-Factors in Bipartite Graphs
Let G be balanced bipart. graph on 2p vertices with min deg δ ≥ p
2.
Let α = δ +
2 . Then G has an ⌊α⌋-regular spanning subgraph. Thm. If G has min deg δ ≥ n/2 + 2
- n/2 · ln(n), then G contains
n/8 edge-disjoint 1-factors.
SLIDE 27 Finding k-Factors in Bipartite Graphs
Let G be balanced bipart. graph on 2p vertices with min deg δ ≥ p
2.
Let α = δ +
2 . Then G has an ⌊α⌋-regular spanning subgraph. Thm. If G has min deg δ ≥ n/2 + 2
- n/2 · ln(n), then G contains
n/8 edge-disjoint 1-factors. Proof. G has bisection H with min deg ≥ δ/2 −
Apply Csaba to get regular subgraph in H.
SLIDE 28 Edge-Disjoint 1-Factors
Thm. If G has n verts, n = pq, q even, with δ(G) ≥ n/2 + q
then G has (n − p)/4 edge-disjoint 1-factors.
SLIDE 29 Edge-Disjoint 1-Factors
Thm. If G has n verts, n = pq, q even, with δ(G) ≥ n/2 + q
then G has (n − p)/4 edge-disjoint 1-factors. Cor. If G has n verts, n even and square, with δ(G) ≥ n/2 + (n ln n)
3 4 ,
then G has n/4 − n/4 edge-disjoint 1-factors.
SLIDE 30 Edge-Disjoint 1-Factors
- Proof. Obtain q parts of size p.
SLIDE 31 Edge-Disjoint 1-Factors
- Proof. Obtain q parts of size p.
Between the parts are bisections of large min deg.
SLIDE 32 Edge-Disjoint 1-Factors
- Proof. Obtain q parts of size p.
Between the parts are bisections of large min deg. Apply Csaba’s thm to split into matchings.
SLIDE 33 Edge-Disjoint 1-Factors
- Proof. Obtain q parts of size p.
Between the parts are bisections of large min deg. Apply Csaba’s thm to split into matchings. Combine into 1-factors by blowing up 1-factors of Kq.
SLIDE 34 General Case: 1-Factors
A similar result holds when n is not (close to) square.
- Lem. For any positive integer n, there exists an integer p ≥ n
such that if q = ⌊n/p⌋, then
◮ q is even, ◮ p − n <
◮ n − q <
◮ n − pq < 4
In particular, a set of size n can be partitioned into parts
- f size p and p + 1 such that there are
at most 2
- 2n1/4 + 5 parts of size p + 1.
SLIDE 35 General Case: 1-Factors
- Cor. Let G be a graph on n vertices, q < n and p = ⌊n/q⌋.
There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg()/q −
- min{deg(), p + 1} · ln(n + q) nbrs in each part.
SLIDE 36 General Case: 1-Factors
- Cor. Let G be a graph on n vertices, q < n and p = ⌊n/q⌋.
There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg()/q −
- min{deg(), p + 1} · ln(n + q) nbrs in each part.
Thus we partition G into ≈ n parts of size p or p + 1. The number of big parts is small: ≤ 2
SLIDE 37 General Case: 1-Factors
- Cor. Let G be a graph on n vertices, q < n and p = ⌊n/q⌋.
There exists a partition of G into q parts of size p or p + 1 such that every vertex has at least deg()/q −
- min{deg(), p + 1} · ln(n + q) nbrs in each part.
Thus we partition G into ≈ n parts of size p or p + 1. The number of big parts is small: ≤ 2
How do we find 1-factors between parts of different size?
SLIDE 38 Generalization of Csaba’s Theorem
- Lem. Let G be bipartite on X ∪ Y with |X| = p + 1 and |Y| = p
such that vertices in X have min deg δ and vertices in Y have min deg δ + 1 for some δ ≥ p/2. Let k ≤ δ+
2δp−p2 2
and let ƒ : V(G) → N satisfy
◮ ƒ() ≤ k for ∈ X, ◮ ƒ() = k for ∈ Y, ◮ and
Then G has an ƒ-factor H. Moreover, H decomposes into matchings that saturate Y.
δ δ + 1 |Y| = p |X| = p + 1 k k k ≤ k ≤ k ≤ k ≤ k
SLIDE 39
Blowing Up
We only blow up 1-factors of Kq that have no edges between parts of size p + 1.
SLIDE 40 Blowing Up
We only blow up 1-factors of Kq that have no edges between parts of size p + 1.
- Lem. Let q be even, and let A be a set in Kq with |A| = ≥ 2.
Then there exist q − 2 + 2 edge-disjoint 1-factors in Kq with no edges inside A.
SLIDE 41 Putting It All T
Partition G into q ≈ n parts of size p and p + 1, p ≈ n, with ≤ 2
- 2n1/4 + 5 parts of size p + 1; call these parts A.
p + 1 p
SLIDE 42 Putting It All T
Partition G into q ≈ n parts of size p and p + 1, p ≈ n, with ≤ 2
- 2n1/4 + 5 parts of size p + 1; call these parts A.
There are ≥ q − 8
- 2n1/4 − 14 edge-disjoint 1-factors of Kq
that use no edges of A.
p + 1 p
SLIDE 43 Putting It All T
Partition G into q ≈ n parts of size p and p + 1, p ≈ n, with ≤ 2
- 2n1/4 + 5 parts of size p + 1; call these parts A.
There are ≥ q − 8
- 2n1/4 − 14 edge-disjoint 1-factors of Kq
that use no edges of A. Blow up using Csaba’s thm and generalization,
- btaining n/4 − O(n7/8 ln(n)) edge-disjoint 1-factors.
p + 1 p
SLIDE 44 Hamiltonian Cycles
- Thm. Let G have min deg δ ≥ n/2 + O(n3/4 ln(n)).
Then G contains n/8 − O(n7/8 ln(n)) edge-disjoint Ham cycles.
SLIDE 45 Hamiltonian Cycles
- Thm. Let G have min deg δ ≥ n/2 + O(n3/4 ln(n)).
Then G contains n/8 − O(n7/8 ln(n)) edge-disjoint Ham cycles. Proof sketch. We partition V(G) into parts of size p and p + 1. There are few parts of size p + 1 (call them set A). Make a set B of size n3/8 of parts of size p, set C the rest.
SLIDE 46 Hamiltonian Cycles in Kq
- Lem. Fix Kq. Let A, B, C be a vertex partition as before.
Then there are q2/2 − 2q7/4 Hamiltonian cycles (not necessarily edge-disjoint) satisfying
◮ every edge of Kq appears in at most q +
◮ no edge within A appears in a cycle, ◮ every edge within B appears in at most q cycles, ◮ every cycle has exactly one edge within B.
SLIDE 47
Blowing Up Hamiltonian Cycles
Blow up each Ham cycle of Kq using 1-factors between parts.
SLIDE 48
Blowing Up Hamiltonian Cycles
Blow up each Ham cycle of Kq using 1-factors between parts. Glue together using the one edge in B.
SLIDE 49 Open Questions
Can the error term in the probabilistic lemma be improved? Can the
Prove the Bollobás–Scott conjecture. What can be proved when the minimum degree is not large? Other uses of our partition theorem?
SLIDE 50 Finding many edge-disjoint Hamiltonian cycles in dense graphs
Stephen G. Hartke
Department of Mathematics University of Nebraska–Lincoln www.math.unl.edu/∼shartke2 hartke@math.unl.edu Joint work with T yler Seacrest
