Neighborhood Unions, Eigenvalues and Disjoint Cycles Paul Horn - - PowerPoint PPT Presentation

Background Results

Neighborhood Unions, Eigenvalues and Disjoint Cycles

Paul Horn

Department of Mathematics and Computer Science Emory University Partially based on joint work with Ron Gould, Emory University and Kazuhide Hirohata, Ibaraki National College of Technology

May 13, 2011

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Basic Question

Many questions in graph theory are of the following type: What simple structural properties of a graph imply that a graph has some more complex property? e.g. ... Dirac (1952): If G has minimum degree n

2, then G is

Hamiltonian. Ore (1960): If the degree-sum of any two non-adjacent vertices in G is at least n, then G is Hamiltonian.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Basic Question

Many questions in graph theory are of the following type: What simple structural properties of a graph imply that a graph has some more complex property? e.g. ... Dirac (1952): If G has minimum degree n

2, then G is

Hamiltonian. Ore (1960): If the degree-sum of any two non-adjacent vertices in G is at least n, then G is Hamiltonian.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Cycles

What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent (pairwise disjoint) cycles.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Cycles

What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent (pairwise disjoint) cycles.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Cycles

What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent (pairwise disjoint) cycles.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Generalizations

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Special case: If |G| is divisible by 3, and G has minimum degree at least 2

3n, then G has a triangle-factor.

A generalization? Hajnal-Szeméredi (1970): If |G| is divisible by k, and G has minimum degree at least k−1

k n, then G has a Kk-factor.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Generalizations

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Special case: If |G| is divisible by 3, and G has minimum degree at least 2

3n, then G has a triangle-factor.

A generalization? Hajnal-Szeméredi (1970): If |G| is divisible by k, and G has minimum degree at least k−1

k n, then G has a Kk-factor.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Generalizations

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Special case: If |G| is divisible by 3, and G has minimum degree at least 2

3n, then G has a triangle-factor.

A generalization? Hajnal-Szeméredi (1970): If |G| is divisible by k, and G has minimum degree at least k−1

k n, then G has a Kk-factor.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Generalizations

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Different degree conditions? Wang (1999): If the degree sum of any two non-adjacent vertices is at least 4k − 1, and |G| ≥ 3k, then G contains k independent cycles. Remarks: Ore-type condition (compared to Dirac-type condition on Corrádi-Hajnal) Strengthens results of Justensen (1989)

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Degree conditions

Dirac-type conditions: Minimum degree condition: Every vertex must have some minimum degree Ore-type conditions: Degree-Sum condition: Vertices may have small degree, degree sum of non-adjacent vertices must be large. Both conditions: Guarantee some ’local expansion’. Dirac: Requires it at every vertex. Ore: Requires it at ’most vertices’.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Degree conditions

Dirac-type conditions: Minimum degree condition: Every vertex must have some minimum degree Ore-type conditions: Degree-Sum condition: Vertices may have small degree, degree sum of non-adjacent vertices must be large. Both conditions: Guarantee some ’local expansion’. Dirac: Requires it at every vertex. Ore: Requires it at ’most vertices’.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Degree conditions

Dirac-type conditions: Minimum degree condition: Every vertex must have some minimum degree Ore-type conditions: Degree-Sum condition: Vertices may have small degree, degree sum of non-adjacent vertices must be large. Split the difference! Neighborhood union condition: Require than the union of the neighborhood of two vertices is large.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Degree conditions

Dirac-type conditions: Minimum degree condition: Every vertex must have some minimum degree Ore-type conditions: Degree-Sum condition: Vertices may have small degree, degree sum of non-adjacent vertices must be large. Neighborhood union condition: Require than the union of the neighborhood of two vertices is large.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Corrádi-Hajnal Revisited

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Gould and J. Faudree (2005): If the neighborhood union of any two non-adjacent vertices in G has size at least 3k, then G contains k independent cycles. Remark: Condition implies |G| ≥ 3k. Best possible? Gould and Faudree conjectured union of 2k + O(1) might suffice. (Union of 2k not sufficient.)

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Corrádi-Hajnal Revisited

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Gould and J. Faudree (2005): If the neighborhood union of any two non-adjacent vertices in G has size at least 3k, then G contains k independent cycles. Remark: Condition implies |G| ≥ 3k. Best possible? Gould and Faudree conjectured union of 2k + O(1) might suffice. (Union of 2k not sufficient.)

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Corrádi-Hajnal Revisited

Corrádi-Hajnal: If G has minimum degree at least 2k and |G| ≥ 3k, then G contains k independent cycles. Theorem (Gould, Hirohata, H.) If the neighborhood union of any two non-adjacent vertices in G has size at least 2k + 1, and |G| ≥ 30k then G contains k independent cycles. Remark: 2k + 1 is best possible, but |G| > 30k is not. Improvement to |G| ≥ 3k is impossible, however: consider an isolated vertex and a K3k−1.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Violates minimality! Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Allows us to swap! Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea:

As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C1, . . . , Ck−1 with |Ck| minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ Gould and Faudree: implicitly used the following lemma: Lemma (Swapping Lemma) Suppose x and y are two non-adjacent leaves in our forest, and the minimal neighborhood union is 3k. Then there exists a cycle Ci, and vertex v ∈ Ci, such that Ci \ {v} ∪ {x} is a cycle, with y ∼ v. Allows them to turn the forest into a path! False with minimal neighborhood union 2k + 1!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ Gould and Faudree: implicitly used the following lemma: Lemma (Swapping Lemma) Suppose x and y are two non-adjacent leaves in our forest, and the minimal neighborhood union is 3k. Then there exists a cycle Ci, and vertex v ∈ Ci, such that Ci \ {v} ∪ {x} is a cycle, with y ∼ v. Allows them to turn the forest into a path! False with minimal neighborhood union 2k + 1!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ We use a more complicated version... Lemma (Swapping Lemma) Suppose x1, x2, and x3 are pairwise disjoint leaves in our cycle. Then there exists i and j with i = j, as well as cycle Ck and z ∈ Ck, such that (Ck − {z}) ∪ {xi} is a cycle and z ∼ xj. Annoyance: We have no guarantee as to which ’swap’ we can make! Savior: More vertices in our forest allow us to sidestep problem.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ Have many vertices (≥ 30k) in G, but want many vertices in forest. Problem: How do we know there are many vertices in our forest? Need to control the size of cycles - key is to control structure between cycles!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

Remarks: ’9’ is best possible. Allows us to prove size of forest is large (≥ 40 vertices).

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Key ideas

Biggest hassle: Use minimality + degree conditions to imply forest is ’nice’ Swapping Lemma + Size of Forest: May assume we have either path of length 8 or two paths of length 4 starting at leaves in our forest all in vertices of degree 2. Final Lemma: Edges between such a structure, and a cycle in

• ur system yields two disjoint cycles!

Completes the proof!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Eigenvalues?

Isn’t there something about eigenvalues in the title? What does this have to do with eigenvalues? Retrospective: To get k independent cycles, suffices to have: Corrádi-Hajnal: Minimum degree 2k. Wang: Minimum degree-sum 4k − 1. Gould, Hirohata, H.: Minimum neighborhood union 2k + 1. The second condition almost subsumes the first, and in many cases is far stronger. Just requires a bit of... expansion. Eigenvalues and expansion are closely related.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Eigenvalues?

Isn’t there something about eigenvalues in the title? What does this have to do with eigenvalues? Retrospective: To get k independent cycles, suffices to have: Corrádi-Hajnal: Minimum degree 2k. Wang: Minimum degree-sum 4k − 1. Gould, Hirohata, H.: Minimum neighborhood union 2k + 1. The second condition almost subsumes the first, and in many cases is far stronger. Just requires a bit of... expansion. Eigenvalues and expansion are closely related.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Eigenvalues?

Isn’t there something about eigenvalues in the title? What does this have to do with eigenvalues? Retrospective: To get k independent cycles, suffices to have: Corrádi-Hajnal: Minimum degree 2k. Wang: Minimum degree-sum 4k − 1. Gould, Hirohata, H.: Minimum neighborhood union 2k + 1. The second condition almost subsumes the first, and in many cases is far stronger. Just requires a bit of... expansion. Eigenvalues and expansion are closely related.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Dense case

Let’s consider dense graphs first: Corrádi-Hajnal: If |G| is divisble by 3, and G has minimum degree 2n

3 , then G has a triangle factor.

Krivelevich, Sudakov and Szabó: If |G| is divisible by 3, and G is d regular with the second largest eigenvalue of the adjacency matrix smaller than cd3/n2 log n for some constant c, then G contains a triangle factor. Remarks: Theorem of KSS may apply even when d is of the order n4/5 log(n)2/5 (much less than 2n

3 ).

Theorem of KSS requires G be regular.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Dense case

Let’s consider dense graphs first: Corrádi-Hajnal: If |G| is divisble by 3, and G has minimum degree 2n

3 , then G has a triangle factor.

Krivelevich, Sudakov and Szabó: If |G| is divisible by 3, and G is d regular with the second largest eigenvalue of the adjacency matrix smaller than cd3/n2 log n for some constant c, then G contains a triangle factor. Remarks: Theorem of KSS may apply even when d is of the order n4/5 log(n)2/5 (much less than 2n

3 ).

Theorem of KSS requires G be regular.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Dense case

Let’s consider dense graphs first: Corrádi-Hajnal: If |G| is divisble by 3, and G has minimum degree 2n

3 , then G has a triangle factor.

Krivelevich, Sudakov and Szabó: If |G| is divisible by 3, and G is d regular with the second largest eigenvalue of the adjacency matrix smaller than cd3/n2 log n for some constant c, then G contains a triangle factor. G much sparser than as required by KSS? Cannot guarantee triangle factor. What can we find? Can we prove something for irregular graphs?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

Since we deal with irregular graphs, the adjacency matrix is not the most appropriate. Underlying matrix: normalized Laplacian L = I − D−1/2AD−1/2 A = adjacency matrix D = diagonal degree matrix Eigenvalues: 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. λ1 > 0 ⇐ ⇒ G connected λn−1 < 2 ⇐ ⇒ no bipartite component. Eigenvalues well suited for use with irregular graphs

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

Since we deal with irregular graphs, the adjacency matrix is not the most appropriate. Underlying matrix: normalized Laplacian L = I − D−1/2AD−1/2 A = adjacency matrix D = diagonal degree matrix Eigenvalues: 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. λ1 > 0 ⇐ ⇒ G connected λn−1 < 2 ⇐ ⇒ no bipartite component. Eigenvalues well suited for use with irregular graphs

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

Since we deal with irregular graphs, the adjacency matrix is not the most appropriate. Underlying matrix: normalized Laplacian L = I − D−1/2AD−1/2 A = adjacency matrix D = diagonal degree matrix Eigenvalues: 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. λ1 > 0 ⇐ ⇒ G connected λn−1 < 2 ⇐ ⇒ no bipartite component. Eigenvalues well suited for use with irregular graphs

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

Since we deal with irregular graphs, the adjacency matrix is not the most appropriate. Underlying matrix: normalized Laplacian L = I − D−1/2AD−1/2 A = adjacency matrix D = diagonal degree matrix Eigenvalues: 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. λ1 > 0 ⇐ ⇒ G connected λn−1 < 2 ⇐ ⇒ no bipartite component. Eigenvalues well suited for use with irregular graphs

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

If L is the normalized Laplacian of G, eigenvalues are 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. σ = max{1 − λ1, λn−1 − 1} is the spectral bound. If σ is small, then we have much control over the structure of

• ur graph!

Lemma (Expander Mixing Lemma) If X, Y ⊆ G and vol(X) =

v∈S deg(v), then

• e(X, Y) − vol(X)vol(Y)

vol(G)

• ≤ σ
• vol(X)vol(Y).

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Spectral Graph Theory:

If L is the normalized Laplacian of G, eigenvalues are 0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2. σ = max{1 − λ1, λn−1 − 1} is the spectral bound. If σ is small, then we have much control over the structure of

• ur graph!

Lemma (Expander Mixing Lemma) If X, Y ⊆ G and vol(X) =

v∈S deg(v), then

• e(X, Y) − vol(X)vol(Y)

vol(G)

• ≤ σ
• vol(X)vol(Y).

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Theorem (H.) Suppose G is a graph with average degree d > 100 and σ <

1

• 10. Then G contains Cσ

√ dn independent cycles, where Cσ is a constant depending only on σ. Remarks: Don’t take d > 100 too seriously. The order of growth √ dn is tight up to the constant Cσ. Condition on average degree as opposed to minimum degree or degree of regularity. Proof: Combines structural tools we used before with spectral tools.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Theorem (H.) Suppose G is a graph with average degree d > 100 and σ <

1

• 10. Then G contains Cσ

√ dn independent cycles, where Cσ is a constant depending only on σ. Remarks: Don’t take d > 100 too seriously. The order of growth √ dn is tight up to the constant Cσ. Condition on average degree as opposed to minimum degree or degree of regularity. Proof: Combines structural tools we used before with spectral tools.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: This time: Let’s assume we have as many cycles as possible, and then that the cycle length is minimized.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: This time: Let’s assume we have as many cycles as possible, and then that the cycle length is minimized. What can we say about our forest, F?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: This time: Let’s assume we have as many cycles as possible, and then that the cycle length is minimized. What can we say about our forest, F? Claim: vol(F) is small! vol(F) ≤ 5 max{σ, 1 √ d }vol(G).

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: Claim: vol(F) ≤ 5 max{σ, 1 √ d }vol(G). Why?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: Claim: vol(F) ≤ 5 max{σ, 1 √ d }vol(G). Why? The expander mixing lemma! Expander Mixing Lemma: For all X, Y ⊆ G

• e(X, Y) − vol(X)vol(Y)

vol(G)

• < σ
• vol(X)vol(Y)

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: Claim: vol(F) ≤ 5 max{σ, 1 √ d }vol(G). Why? The expander mixing lemma! If e(F) ≥ n, then F contains a cycle.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea

Let’s start as before, with a cycle system plus a forest: Most edges are not incident to the forest. How do we use this to show there are many cycles? Let’s first consider for regular graphs.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. So what?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. So what?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. So what?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. So what?

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. This will be a contradiction of regularity of ℓ is too small (say ℓ < √n)! But this argument requires regularity and doesn’t quite get C √ dn cycles.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof Idea (Regular Graphs):

Suppose G is d-regular. We start out with dn

2 edges. Toss out all edges incident to

F and still have ≈ dn

2 edges.

Minimality ⇒ no chords, thus have ≈ dn

2 edges between

cycles. If ℓ cycles in our system, must have ≈

dn 2(ℓ

2) ≈ dn

ℓ2 edges

some pair of cycles. We need a more sophisticated approach.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

Problem: No regularity to contradict!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

If two cycles in our system, are connected by more than 9 edges, then one vertex dominates almost all of them. Idea: Create a helper graph H, where the vertices are cycles C1, . . . , Cℓ. Place an edge between cycles if they are connected by more than 9 edges. Caveat: If the same vertex in Ci has high degree to both Cx and Cy, place only one edge.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

If two cycles in our system, are connected by more than 9 edges, then one vertex dominates almost all of them. Idea: Create a helper graph H, where the vertices are cycles C1, . . . , Cℓ. Place an edge between cycles if they are connected by more than 9 edges. Caveat: If the same vertex in Ci has high degree to both Cx and Cy, place only one edge.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

If two cycles in our system, are connected by more than 9 edges, then one vertex dominates almost all of them. Idea: Create a helper graph H, where the vertices are cycles C1, . . . , Cℓ. Place an edge between cycles if they are connected by more than 9 edges. Caveat: If the same vertex in Ci has high degree to both Cx and Cy, place only one edge.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

Claim: H must be acyclic. Proof: Otherwise minimality contradicted. So what? Implies that most of the degree is concentrated in few (≤ ℓ − 1) vertices.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

Claim: H must be acyclic. Proof: Otherwise minimality contradicted. So what? Implies that most of the degree is concentrated in few (≤ ℓ − 1) vertices.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

To finish: Aim for one of two contradictions.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

To finish: Aim for one of two contradictions. Either: There is such a high concentration of edges in so few vertices that it contradicts the expander mixing lemma if ℓ < C √ dn. Or not enough edges can be accounted for if ℓ < C √ dn.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Proof idea: Irregular Graphs

We still want to use Lemma Suppose C1 and C2 are two cycles with |C1| + |C2| ≥ 7, and e(C1, C2) edges between them. If e(C1, C2) ≥ 9, all but at most

• ne edge between cycles are dominated by a single vertex.

To finish: Aim for one of two contradictions. Either: There is such a high concentration of edges in so few vertices that it contradicts the expander mixing lemma if ℓ < C √ dn. Or not enough edges can be accounted for if ℓ < C √ dn. Either way: We’re happy!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Optimality

Consider the graph G(n, k) on (k + 1)n vertices, consisting of a Kn with k independent vertices to each vertex of the Kn. G(4, 2) For k = ǫn: Average Degree: d ≈ 1+2ǫ

ǫ

# Cycles: ⌊ n

3⌋

# Vertices: N ≈ ǫn2 1 3 √ dN = 1 3

• (1 + 2ǫ)

ǫ ǫn2 > n 3 Result is of right order.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Optimality

Consider the graph G(n, k) on (k + 1)n vertices, consisting of a Kn with k independent vertices to each vertex of the Kn. G(4, 2) For k = ǫn: Average Degree: d ≈ 1+2ǫ

ǫ

# Cycles: ⌊ n

3⌋

# Vertices: N ≈ ǫn2 1 3 √ dN = 1 3

• (1 + 2ǫ)

ǫ ǫn2 > n 3 Result is of right order.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Optimality

Consider the graph G(n, k) on (k + 1)n vertices, consisting of a Kn with k independent vertices to each vertex of the Kn. G(4, 2) For k = ǫn: Average Degree: d ≈ 1+2ǫ

ǫ

# Cycles: ⌊ n

3⌋

# Vertices: N ≈ ǫn2 1 3 √ dN = 1 3

• (1 + 2ǫ)

ǫ ǫn2 > n 3 Result is of right order.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Optimality

Consider the graph G(n, k) on (k + 1)n vertices, consisting of a Kn with k independent vertices to each vertex of the Kn. G(4, 2) For k = ǫn: Average Degree: d ≈ 1+2ǫ

ǫ

# Cycles: ⌊ n

3⌋

# Vertices: N ≈ ǫn2 1 3 √ dN = 1 3

• (1 + 2ǫ)

ǫ ǫn2 > n 3 Result is of right order.

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Concluding Remarks

General techniques are quite applicable: With Gould, Hirohata established neighborhood union results for chorded cycles. Spectral techniques applicable to many families: k-chorded cycles, etc. Question: Can sharp constant be obtained by spectral techniques?

Thank You!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles

Background Results

Concluding Remarks

General techniques are quite applicable: With Gould, Hirohata established neighborhood union results for chorded cycles. Spectral techniques applicable to many families: k-chorded cycles, etc. Question: Can sharp constant be obtained by spectral techniques?

Thank You!

Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles