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
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?
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.
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 ⌊ 5 n/ 224 ⌋ edge-disjoint Hamiltonian cycles. This gives roughly n/ 46 edge-disjoint Hamiltonian cycles, which gives n/ 23 edge-disjoint 1-factors.
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.
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.
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. Thm. [H, Seacrest 2011+] Let G have min deg δ ≥ n/ 2 + O ( n 3 / 4 ln ( n )) . Then G contains n/ 8 − O ( n 7 / 8 ln ( n )) edge-disjoint Ham cycles. Our proof avoids the Regularity Lemma and hence the constants are much smaller.
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. Thm. [H, Seacrest 2011+] Let G have min deg δ ≥ n/ 2 + O ( n 3 / 4 ln ( n )) . Then G contains n/ 8 − O ( n 7 / 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 .)
Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H with at least half the edges of G .
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 deg H ( ) < deg G ( ) / 2 , then push to the other part.
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 deg H ( ) < deg G ( ) / 2 , then push to the other part. Repeat; the number of edges in H strictly increases. �
Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) ≥ deg G ( ) / 2 for all vertices . Proof. Partition verts of G ; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( ) / 2 , then push to the other part. Repeat; the number of edges in H strictly increases. �
Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) ≥ deg G ( ) / 2 for all vertices . Proof. Partition verts of G ; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( ) / 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 . �
Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) ≥ deg G ( ) / 2 for all vertices . Proof. Partition verts of G ; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( ) / 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.
Large Bipartite Subgraphs Prop. Every graph G has a bipartite subgraph H where deg H ( ) ≥ deg G ( ) / 2 for all vertices . Proof. Partition verts of G ; let H be the induced bipartite subgraph. If has deg H ( ) < deg G ( ) / 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.
Large Bipartite Subgraphs Can both properties be simultaneously obtained?
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 deg H ( ) ≥ ⌊ deg G ( ) / 2 ⌋ for all .
Sharpness The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with deg H ( ) ≥ ⌊ deg G ( ) / 2 ⌋ + 1 for all . Let k < n/ 2 . K k n − k K k ∨ ( n − k ) K 1
Sharpness The conjecture would be sharp: There exists a graph G with an even number of vertices with no bisection H with deg H ( ) ≥ ⌊ deg G ( ) / 2 ⌋ + 1 for all . Let k < n/ 2 . K k For any partition, both n − k partite sets have vertices from the independent set. K k ∨ ( n − k ) K 1
Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices , 1 � deg H ( ) ≥ deg G ( ) − deg G ( ) ln ( n ) . 2 [similar result by A. Bush 2009]
Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices , 1 � deg H ( ) ≥ deg G ( ) − deg G ( ) ln ( n ) . 2 [similar result by A. Bush 2009] Proof Outline. Arbitrarily pair the vertices. Randomly split each pair between the two partite sets.
Probabilistic Approach Thm. Any graph G on n (even) vertices has a bisection H such that for all vertices , 1 � deg H ( ) ≥ deg G ( ) − deg G ( ) ln ( n ) . 2 [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.
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.
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 � n/q 3 neighbors in P . 3
Finding k -Factors in Bipartite Graphs Thm. [Csaba 2007] Let G be balanced bipart. graph on 2 p vertices with min deg δ ≥ p 2 . � δ + p ( 2 δ − p ) Let α = . 2 Then G has an ⌊ α ⌋ -regular spanning subgraph.
Finding k -Factors in Bipartite Graphs Thm. [Csaba 2007] Let G be balanced bipart. graph on 2 p vertices with min deg δ ≥ p 2 . � δ + p ( 2 δ − p ) Let α = . 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.
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