gregory j. puleo
TRANSCRIPT
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On a Conjecture of Erdos, Gallai, and Tuza
Gregory J. Puleo
Coordinated Science Lab, UIUC
UIUC Graph Theory SeminarSeptember 23, 2014
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How many edges in a triangle-free graph?
Definition
A graph is triangle-free if it has no subgraph isomorphic to K3.
Theorem (Mantel 1907)
If G is a triangle-free n-vertex graph, then G has at most n2/4 edges.Equality holds if and only if G = Kn/2,n/2.
Theorem (Turan 1941)
If G is an n-vertex graph with no copy of Kr , then G has at most(r−2r−1
)n2
2 edges.
Equality holds if and only if G = Kn/(r−1),··· ,n/(r−1).
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How many edges in a triangle-free graph?
Definition
A graph is triangle-free if it has no subgraph isomorphic to K3.
Theorem (Mantel 1907)
If G is a triangle-free n-vertex graph, then G has at most n2/4 edges.Equality holds if and only if G = Kn/2,n/2.
Theorem (Turan 1941)
If G is an n-vertex graph with no copy of Kr , then G has at most(r−2r−1
)n2
2 edges.
Equality holds if and only if G = Kn/(r−1),··· ,n/(r−1).
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How hard is it to make a graph bipartite?
Theorem (Erdos 1965)
Any graph G can be made bipartite by removing at most |E (G )| /2 edges.
Proof.
Randomly color each vertex red or blue, each with probability 1/2.Delete all monochromatic edges.
Each edge has probability 1/2 of being deleted, so the expectednumber of deleted edges is |E (G )| /2.
Corollary
If G has n vertices, then G can be made bipartite by deletingat most n2/4 edges. The worst-case graph is Kn.
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How hard is it to make a graph bipartite?
Theorem (Erdos 1965)
Any graph G can be made bipartite by removing at most |E (G )| /2 edges.
Proof.
Randomly color each vertex red or blue, each with probability 1/2.Delete all monochromatic edges.
Each edge has probability 1/2 of being deleted, so the expectednumber of deleted edges is |E (G )| /2.
Corollary
If G has n vertices, then G can be made bipartite by deletingat most n2/4 edges. The worst-case graph is Kn.
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How hard is it to make a graph bipartite?
Theorem (Erdos 1965)
Any graph G can be made bipartite by removing at most |E (G )| /2 edges.
Proof.
Randomly color each vertex red or blue, each with probability 1/2.Delete all monochromatic edges.
Each edge has probability 1/2 of being deleted, so the expectednumber of deleted edges is |E (G )| /2.
Corollary
If G has n vertices, then G can be made bipartite by deletingat most n2/4 edges. The worst-case graph is Kn.
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How hard is it to make a graph bipartite?
Theorem (Erdos 1965)
Any graph G can be made bipartite by removing at most |E (G )| /2 edges.
Proof.
Randomly color each vertex red or blue, each with probability 1/2.Delete all monochromatic edges.
Each edge has probability 1/2 of being deleted, so the expectednumber of deleted edges is |E (G )| /2.
Corollary
If G has n vertices, then G can be made bipartite by deletingat most n2/4 edges. The worst-case graph is Kn.
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Hitting sets and triangle-independent sets
Definition
X ⊂ E (G ) is a hitting set if it contains at least one edgefrom each triangle of G .
A ⊂ E (G ) is triangle-independent if it contains at most one edgefrom each triangle of G .
Definition
τ1(G ) is the smallest size of a hitting set in G .
α1(G ) is the largest size of a triangle-independent set in G .
The Erdos bound implies: τ1(G ) ≤ n2/4.Mantel’s Theorem implies: α1(G ) ≤ n2/4.
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Hitting sets and triangle-independent sets
Definition
X ⊂ E (G ) is a hitting set if it contains at least one edgefrom each triangle of G .
A ⊂ E (G ) is triangle-independent if it contains at most one edgefrom each triangle of G .
Definition
τ1(G ) is the smallest size of a hitting set in G .
α1(G ) is the largest size of a triangle-independent set in G .
The Erdos bound implies: τ1(G ) ≤ n2/4.Mantel’s Theorem implies: α1(G ) ≤ n2/4.
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Hitting sets and triangle-independent sets
Definition
X ⊂ E (G ) is a hitting set if it contains at least one edgefrom each triangle of G .
A ⊂ E (G ) is triangle-independent if it contains at most one edgefrom each triangle of G .
Definition
τ1(G ) is the smallest size of a hitting set in G .
α1(G ) is the largest size of a triangle-independent set in G .
The Erdos bound implies: τ1(G ) ≤ n2/4.Mantel’s Theorem implies: α1(G ) ≤ n2/4.
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The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
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The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 13: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/13.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 14: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/14.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 15: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/15.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 16: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/16.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 17: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/17.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 18: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/18.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 19: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/19.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
![Page 20: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/20.jpg)
The Erdos–Gallai–Tuza Conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
The conjecture is sharp, if true:
Consider the complete graph Kn, where n is even:
τ1(Kn) =(n2
)− n2/4 = n2/4− n/2,
α1(Kn) = n/2,
Thus α1(Kn) + τ1(Kn) = n2/4.
Next, consider the complete bipartite graph Kn/2,n/2:
τ1(Kn/2,n/2) = 0,
α1(Kn/2,n/2) = n2/4,
Thus α1(Kn/2,n/2) + τ1(Kn/2,n/2) = n2/4.
More generally, the conjecture is sharp for Kr1,r1 ∨ · · · ∨ Krt ,rt .
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From τ1 to τB
Definition
τ1(G ) is the smallest size of a hitting set in G .
Equivalent Definition
τ1(G ) is the smallest size of an edge set X such that G −X is triangle-free.
Definition
τB(G ) is the smallest size of an edge set X such that G − X is bipartite.
Observe that τ1(G ) ≤ τB(G ) ≤ n2/4.
Conjecture
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ n2/4.
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From τ1 to τB
Definition
τ1(G ) is the smallest size of a hitting set in G .
Equivalent Definition
τ1(G ) is the smallest size of an edge set X such that G −X is triangle-free.
Definition
τB(G ) is the smallest size of an edge set X such that G − X is bipartite.
Observe that τ1(G ) ≤ τB(G ) ≤ n2/4.
Conjecture
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ n2/4.
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From τ1 to τB
Definition
τ1(G ) is the smallest size of a hitting set in G .
Equivalent Definition
τ1(G ) is the smallest size of an edge set X such that G −X is triangle-free.
Definition
τB(G ) is the smallest size of an edge set X such that G − X is bipartite.
Observe that τ1(G ) ≤ τB(G ) ≤ n2/4.
Conjecture
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ n2/4.
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From τ1 to τB
Definition
τ1(G ) is the smallest size of a hitting set in G .
Equivalent Definition
τ1(G ) is the smallest size of an edge set X such that G −X is triangle-free.
Definition
τB(G ) is the smallest size of an edge set X such that G − X is bipartite.
Observe that τ1(G ) ≤ τB(G ) ≤ n2/4.
Conjecture
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ n2/4.
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Two approaches to the conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Two obvious ways to try to find partial results:
Find c > 1/4 such that α1(G ) + τ1(G ) ≤ cn2 for all G .Find a class F such that α1(G ) + τ1(G ) ≤ n2/4 for all G ∈ F .
Theorem (P. 2014+)
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ 5n2/16.
Theorem (P. 2014+)
If G has no induced K−4 , then α1(G ) + τB(G ) ≤ n2/4.
In the rest of the talk, we prove these and other results.K−4
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Two approaches to the conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Two obvious ways to try to find partial results:
Find c > 1/4 such that α1(G ) + τ1(G ) ≤ cn2 for all G .Find a class F such that α1(G ) + τ1(G ) ≤ n2/4 for all G ∈ F .
Theorem (P. 2014+)
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ 5n2/16.
Theorem (P. 2014+)
If G has no induced K−4 , then α1(G ) + τB(G ) ≤ n2/4.
In the rest of the talk, we prove these and other results.K−4
![Page 27: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/27.jpg)
Two approaches to the conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Two obvious ways to try to find partial results:
Find c > 1/4 such that α1(G ) + τ1(G ) ≤ cn2 for all G .Find a class F such that α1(G ) + τ1(G ) ≤ n2/4 for all G ∈ F .
Theorem (P. 2014+)
If G is an n-vertex graph, then α1(G ) + τB(G ) ≤ 5n2/16.
Theorem (P. 2014+)
If G has no induced K−4 , then α1(G ) + τB(G ) ≤ n2/4.
In the rest of the talk, we prove these and other results.K−4
![Page 28: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/28.jpg)
Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.
Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
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Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
![Page 30: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/30.jpg)
Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
![Page 31: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/31.jpg)
Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
![Page 32: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/32.jpg)
Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
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Two Lemmas
u v
Let b(G ) = max{|B| : G [B] is bipartite}.
Lemma
If A is triangle-independent, then for any uv ∈ A,the induced subgraph G [NA(u) ∪ NA(v)] isbipartite with dA(u) + dA(v) vertices.Thus, dA(u) + dA(v) ≤ b(G ).
Lemma
If A is triangle-independent, then |A| ≤ nb(G)4.
Proof.
For any uv ∈ A, the first lemma gives dA(u) + dA(v) ≤ b(G ).
Summing over all edges in A, we get∑
v dA(v)2 ≤ |A| b(G ).
Cauchy-Schwarz gives∑
v dA(v)2 ≥ 4|A|2n , and the conclusion follows.
![Page 34: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/34.jpg)
α1(G ) + τB(G ) ≤ 5n2/16
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is a graph, and B is a vertex set such that G [B] is bipartite, then
τB(G ) ≤ 12
(∣∣∣E (G ) \(B2
)∣∣∣).
Theorem
For any n-vertex graph G , α1(G ) + τB(G ) ≤ 5n2/16.
Proof.
EFPS yields τB(G ) ≤ 12
[(n2
)−(b(G)
2
)]≤ n2
4 −b(G)2
4 .
Thus, α1(G ) + τB(G ) ≤ nb(G)4 + n2
4 −b(G)2
4 .The right side is maximized when b(G ) = n/2, yieldingα1(G ) + τB(G ) ≤ 5n2/16.
![Page 35: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/35.jpg)
α1(G ) + τB(G ) ≤ 5n2/16
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is a graph, and B is a vertex set such that G [B] is bipartite, then
τB(G ) ≤ 12
(∣∣∣E (G ) \(B2
)∣∣∣).Theorem
For any n-vertex graph G , α1(G ) + τB(G ) ≤ 5n2/16.
Proof.
EFPS yields τB(G ) ≤ 12
[(n2
)−(b(G)
2
)]≤ n2
4 −b(G)2
4 .
Thus, α1(G ) + τB(G ) ≤ nb(G)4 + n2
4 −b(G)2
4 .The right side is maximized when b(G ) = n/2, yieldingα1(G ) + τB(G ) ≤ 5n2/16.
![Page 36: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/36.jpg)
α1(G ) + τB(G ) ≤ 5n2/16
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is a graph, and B is a vertex set such that G [B] is bipartite, then
τB(G ) ≤ 12
(∣∣∣E (G ) \(B2
)∣∣∣).Theorem
For any n-vertex graph G , α1(G ) + τB(G ) ≤ 5n2/16.
Proof.
EFPS yields τB(G ) ≤ 12
[(n2
)−(b(G)
2
)]≤ n2
4 −b(G)2
4 .
Thus, α1(G ) + τB(G ) ≤ nb(G)4 + n2
4 −b(G)2
4 .The right side is maximized when b(G ) = n/2, yieldingα1(G ) + τB(G ) ≤ 5n2/16.
![Page 37: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/37.jpg)
α1(G ) + τB(G ) ≤ 5n2/16
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is a graph, and B is a vertex set such that G [B] is bipartite, then
τB(G ) ≤ 12
(∣∣∣E (G ) \(B2
)∣∣∣).Theorem
For any n-vertex graph G , α1(G ) + τB(G ) ≤ 5n2/16.
Proof.
EFPS yields τB(G ) ≤ 12
[(n2
)−(b(G)
2
)]≤ n2
4 −b(G)2
4 .
Thus, α1(G ) + τB(G ) ≤ nb(G)4 + n2
4 −b(G)2
4 .
The right side is maximized when b(G ) = n/2, yieldingα1(G ) + τB(G ) ≤ 5n2/16.
![Page 38: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/38.jpg)
α1(G ) + τB(G ) ≤ 5n2/16
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is a graph, and B is a vertex set such that G [B] is bipartite, then
τB(G ) ≤ 12
(∣∣∣E (G ) \(B2
)∣∣∣).Theorem
For any n-vertex graph G , α1(G ) + τB(G ) ≤ 5n2/16.
Proof.
EFPS yields τB(G ) ≤ 12
[(n2
)−(b(G)
2
)]≤ n2
4 −b(G)2
4 .
Thus, α1(G ) + τB(G ) ≤ nb(G)4 + n2
4 −b(G)2
4 .The right side is maximized when b(G ) = n/2, yieldingα1(G ) + τB(G ) ≤ 5n2/16.
![Page 39: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/39.jpg)
Two versions of the conjecture
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Definition
A graph is triangular if each of its edges lies in some triangle.
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex triangular graph, then α1(G ) + τ1(G ) ≤ n2/4.
Later formulations of the conjecture, by both Erdos and Tuza, silentlydropped the “triangular” condition.
Question (Grinberg 2012)
Are these two forms of the conjecture equivalent?
Answer (P. 2014+): YES.
![Page 40: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/40.jpg)
Two versions of the conjecture
Conjecture (Erdos–Gallai–Tuza 1996 ...sort of)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Definition
A graph is triangular if each of its edges lies in some triangle.
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex triangular graph, then α1(G ) + τ1(G ) ≤ n2/4.
Later formulations of the conjecture, by both Erdos and Tuza, silentlydropped the “triangular” condition.
Question (Grinberg 2012)
Are these two forms of the conjecture equivalent?
Answer (P. 2014+): YES.
![Page 41: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/41.jpg)
Two versions of the conjecture
Conjecture (Erdos–Gallai–Tuza 1996 ...sort of)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Definition
A graph is triangular if each of its edges lies in some triangle.
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex triangular graph, then α1(G ) + τ1(G ) ≤ n2/4.
Later formulations of the conjecture, by both Erdos and Tuza, silentlydropped the “triangular” condition.
Question (Grinberg 2012)
Are these two forms of the conjecture equivalent?
Answer (P. 2014+): YES.
![Page 42: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/42.jpg)
Two versions of the conjecture
Conjecture (Erdos–Gallai–Tuza 1996 ...sort of)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Definition
A graph is triangular if each of its edges lies in some triangle.
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex triangular graph, then α1(G ) + τ1(G ) ≤ n2/4.
Later formulations of the conjecture, by both Erdos and Tuza, silentlydropped the “triangular” condition.
Question (Grinberg 2012)
Are these two forms of the conjecture equivalent?
Answer (P. 2014+): YES.
![Page 43: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/43.jpg)
Two versions of the conjecture
Conjecture (Erdos–Gallai–Tuza 1996 ...sort of)
If G is an n-vertex graph, then α1(G ) + τ1(G ) ≤ n2/4.
Definition
A graph is triangular if each of its edges lies in some triangle.
Conjecture (Erdos–Gallai–Tuza 1996)
If G is an n-vertex triangular graph, then α1(G ) + τ1(G ) ≤ n2/4.
Later formulations of the conjecture, by both Erdos and Tuza, silentlydropped the “triangular” condition.
Question (Grinberg 2012)
Are these two forms of the conjecture equivalent?
Answer (P. 2014+): YES.
![Page 44: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/44.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.Let X = E (G )− A.Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S ,S ].
![Page 45: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/45.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.
Let X = E (G )− A.Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S ,S ].
![Page 46: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/46.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.Let X = E (G )− A.
Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S ,S ].
![Page 47: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/47.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.Let X = E (G )− A.Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .
Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S ,S ].
![Page 48: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/48.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.Let X = E (G )− A.Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.
Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S ,S ].
![Page 49: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/49.jpg)
Warmup: α1(G ) + τ1(G ) ≤ |E (G )|
Theorem (Erdos–Gallai–Tuza 1996)
In any graph G , α1(G ) + τ1(G ) ≤ |E (G )|.
Proof.
Let A be a largest triangle-independent set, so that α1(G ) = |A|.Let X = E (G )− A.Any triangle of G contains at most one edge of A, thus contains at leasttwo edges of X .Thus, X is a hitting set (and then some!), so that τ1(G ) ≤ |X |.Therefore α1(G ) + τ1(G ) ≤ |A|+ |X | = |E (G )|.
This proof is global, dealing with all edges of G .We localize it, dealing only with the edges of some edge cut [S , S ].
![Page 50: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/50.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S , S ]− C ); now X hits all triangles.
![Page 51: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/51.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .
Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S ,S ]− C ); now X hits all triangles.
![Page 52: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/52.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.
Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S ,S ]− C ); now X hits all triangles.
![Page 53: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/53.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].
Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S ,S ]− C ); now X hits all triangles.
![Page 54: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/54.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .
X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S ,S ]− C ); now X hits all triangles.
![Page 55: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/55.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .
Let X = X1 ∪ X2 ∪ ([S ,S ]− C ); now X hits all triangles.
![Page 56: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/56.jpg)
Edge cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
S S
Proof.
Let A be any triangle-independent set in G .Let X1, X2 be smallest hitting sets in G [S ], G [S ] respectively.Let A1 = A ∩ E (G [S ]), A2 = A ∩ E (G [S ]), C = A ∩ [S ,S ].Have |A1|+ |X1| ≤ α1(G [S ]) + τ1(G [S ]) and likewise for S .X1 ∪ X2 hits all triangles except those with vertices in both S and S .Let X = X1 ∪ X2 ∪ ([S , S ]− C ); now X hits all triangles.
![Page 57: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/57.jpg)
Minimum Counterexamples have Dense Cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Proof.
Lemma + minimality gives
α1(G ) + τ1(G ) ≤ |S |2 /4 + (n − |S |)2/4 +∣∣[S , S ]
∣∣
= n2/4− |S | (n − |S |)/2 +∣∣[S ,S ]
∣∣ .
Since α1(G ) + τ1(G ) > n2/4, the claim follows.
![Page 58: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/58.jpg)
Minimum Counterexamples have Dense Cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Proof.
Lemma + minimality gives
α1(G ) + τ1(G ) ≤ |S |2 /4 + (n − |S |)2/4 +∣∣[S , S ]
∣∣
= n2/4− |S | (n − |S |)/2 +∣∣[S ,S ]
∣∣ .
Since α1(G ) + τ1(G ) > n2/4, the claim follows.
![Page 59: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/59.jpg)
Minimum Counterexamples have Dense Cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Proof.
Lemma + minimality gives
α1(G ) + τ1(G ) ≤ |S |2 /4 + (n − |S |)2/4 +∣∣[S , S ]
∣∣
= n2/4− |S | (n − |S |)/2 +∣∣[S , S ]
∣∣ .Since α1(G ) + τ1(G ) > n2/4, the claim follows.
![Page 60: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/60.jpg)
Minimum Counterexamples have Dense Cuts
Lemma (P. 2014+)
For any S ⊂ V (G ),α1(G ) + τ1(G ) ≤ α1(G [S ]) + τ1(G [S ]) + α1(G [S ]) + τ1(G [S ]) +
∣∣[S , S ]∣∣ .
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Proof.
Lemma + minimality gives
α1(G ) + τ1(G ) ≤ |S |2 /4 + (n − |S |)2/4 +∣∣[S , S ]
∣∣= n2/4− |S | (n − |S |)/2 +
∣∣[S , S ]∣∣ .
Since α1(G ) + τ1(G ) > n2/4, the claim follows.
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Minimum Degree in Minimum Counterexamples
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Improvement
If G is a vertex-minimal counterexample, then δ(G ) > n/2.
Thus, G is triangular.
Corollary
If α1(G ) + τ1(G ) ≤ n2/4 for all triangular graphs G , thenα1(G ) + τ1(G ) ≤ n2/4 for all graphs G .
That is, the two forms of the conjecture are equivalent.
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Minimum Degree in Minimum Counterexamples
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Improvement
If G is a vertex-minimal counterexample, then δ(G ) > n/2.Thus, G is triangular.
Corollary
If α1(G ) + τ1(G ) ≤ n2/4 for all triangular graphs G , thenα1(G ) + τ1(G ) ≤ n2/4 for all graphs G .
That is, the two forms of the conjecture are equivalent.
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Minimum Degree in Minimum Counterexamples
Theorem
If G is a vertex-minimal graph such that α1(G ) + τ1(G ) > n2/4, then forall proper nonempty S ⊂ V (G ), we have
∣∣[S ,S ]∣∣ > |S | (n − |S |)/2.
Improvement
If G is a vertex-minimal counterexample, then δ(G ) > n/2.Thus, G is triangular.
Corollary
If α1(G ) + τ1(G ) ≤ n2/4 for all triangular graphs G , thenα1(G ) + τ1(G ) ≤ n2/4 for all graphs G .
That is, the two forms of the conjecture are equivalent.
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Dense Cuts for τB
Lemma
For any triangle-independent A and any S ⊂ V (G ),α1(G ) + τB(G ) ≤ α1(G [S ]) + τB(G [S ]) + α1(G [S ]) + τB(G [S ])+
12
∣∣[S ,S ]∣∣+∣∣A ∩ [S ,S ]
∣∣ .
Proof.
Suffices to show that τB(G ) ≤ τB(G [S ]) + τB(G [S ]) + 12
∣∣[S ,S ]∣∣.
Given bipartitions of S and S , consider the two ways to combine themyielding a bipartition of V (G ).
S S
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Dense Cuts for τB
Lemma
For any triangle-independent A and any S ⊂ V (G ),α1(G ) + τB(G ) ≤ α1(G [S ]) + τB(G [S ]) + α1(G [S ]) + τB(G [S ])+
12
∣∣[S ,S ]∣∣+∣∣A ∩ [S ,S ]
∣∣ .Proof.
Suffices to show that τB(G ) ≤ τB(G [S ]) + τB(G [S ]) + 12
∣∣[S ,S ]∣∣.
Given bipartitions of S and S , consider the two ways to combine themyielding a bipartition of V (G ).
S S
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Dense Cuts for τB
Lemma
For any triangle-independent A and any S ⊂ V (G ),α1(G ) + τB(G ) ≤ α1(G [S ]) + τB(G [S ]) + α1(G [S ]) + τB(G [S ])+
12
∣∣[S ,S ]∣∣+∣∣A ∩ [S ,S ]
∣∣ .Proof.
Suffices to show that τB(G ) ≤ τB(G [S ]) + τB(G [S ]) + 12
∣∣[S ,S ]∣∣.
Given bipartitions of S and S , consider the two ways to combine themyielding a bipartition of V (G ).
S S
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Dense Cuts for τB
Lemma
For any triangle-independent A and any S ⊂ V (G ),α1(G ) + τB(G ) ≤ α1(G [S ]) + τB(G [S ]) + α1(G [S ]) + τB(G [S ])+
12
∣∣[S ,S ]∣∣+∣∣A ∩ [S ,S ]
∣∣ .Proof.
Suffices to show that τB(G ) ≤ τB(G [S ]) + τB(G [S ]) + 12
∣∣[S ,S ]∣∣.
Given bipartitions of S and S , consider the two ways to combine themyielding a bipartition of V (G ).
S S
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Clique Cuts
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4 and A istriangle-independent, then for any nonempty proper S ⊂ V (G ), we have12
∣∣[S , S ]∣∣+∣∣A ∩ [S , S ]
∣∣ > |S | (n − |S |)/2.
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4, if A istriangle-independent, and if S is a clique, then
∣∣[S ,S ]∣∣ > (|S |− 2)(n−|S |).
Proof.
Since S is a clique, each vertex v ∈ S has at most one A-neighbor in S .Thus,
∣∣A ∩ [S ,S ]∣∣ ≤ n − |S |.
S vcan’t happen:
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Clique Cuts
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4 and A istriangle-independent, then for any nonempty proper S ⊂ V (G ), we have12
∣∣[S , S ]∣∣+∣∣A ∩ [S , S ]
∣∣ > |S | (n − |S |)/2.
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4, if A istriangle-independent, and if S is a clique, then
∣∣[S , S ]∣∣ > (|S |− 2)(n−|S |).
Proof.
Since S is a clique, each vertex v ∈ S has at most one A-neighbor in S .Thus,
∣∣A ∩ [S ,S ]∣∣ ≤ n − |S |.
S vcan’t happen:
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Clique Cuts
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4 and A istriangle-independent, then for any nonempty proper S ⊂ V (G ), we have12
∣∣[S , S ]∣∣+∣∣A ∩ [S , S ]
∣∣ > |S | (n − |S |)/2.
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4, if A istriangle-independent, and if S is a clique, then
∣∣[S , S ]∣∣ > (|S |− 2)(n−|S |).
Proof.
Since S is a clique, each vertex v ∈ S has at most one A-neighbor in S .Thus,
∣∣A ∩ [S , S ]∣∣ ≤ n − |S |.
S vcan’t happen:
![Page 71: Gregory J. Puleo](https://reader030.vdocuments.us/reader030/viewer/2022012705/61a6c0565b6f8465305c6f88/html5/thumbnails/71.jpg)
Clique Cuts
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4 and A istriangle-independent, then for any nonempty proper S ⊂ V (G ), we have12
∣∣[S , S ]∣∣+∣∣A ∩ [S , S ]
∣∣ > |S | (n − |S |)/2.
Corollary
If G is a vertex-minimal graph with α1(G ) + τB(G ) > n2/4, if A istriangle-independent, and if S is a clique, then
∣∣[S , S ]∣∣ > (|S |− 2)(n−|S |).
Proof.
Since S is a clique, each vertex v ∈ S has at most one A-neighbor in S .Thus,
∣∣A ∩ [S , S ]∣∣ ≤ n − |S |.
S vcan’t happen:
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.Let S be a maximum clique, so that |S | ≥ 3.Corollary gives
∣∣[S ,S ]∣∣ > (|S | − 2)(n − |S |), so some v ∈ S has
dS(v) ≥ |S | − 1.Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
Sv
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.
Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.Let S be a maximum clique, so that |S | ≥ 3.Corollary gives
∣∣[S ,S ]∣∣ > (|S | − 2)(n − |S |), so some v ∈ S has
dS(v) ≥ |S | − 1.Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
Sv
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.
Let S be a maximum clique, so that |S | ≥ 3.Corollary gives
∣∣[S ,S ]∣∣ > (|S | − 2)(n − |S |), so some v ∈ S has
dS(v) ≥ |S | − 1.Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
Sv
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.Let S be a maximum clique, so that |S | ≥ 3.
Corollary gives∣∣[S ,S ]
∣∣ > (|S | − 2)(n − |S |), so some v ∈ S hasdS(v) ≥ |S | − 1.Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
S
v
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.Let S be a maximum clique, so that |S | ≥ 3.Corollary gives
∣∣[S ,S ]∣∣ > (|S | − 2)(n − |S |), so some v ∈ S has
dS(v) ≥ |S | − 1.
Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
Sv
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K−4 -free Graphs
Theorem
If G is K−4 -free, then α1(G ) + τB(G ) ≤ n2/4.
Proof (by contradiction).
First, show that α1(G ) + τB(G ) ≤ n2/4 when G is triangle-free.Among the K−4 -free graphs with α1(G ) + τB(G ) > n2/4, take Gvertex-minimal. Note that G is vertex-minimal among all graphs.Let S be a maximum clique, so that |S | ≥ 3.Corollary gives
∣∣[S ,S ]∣∣ > (|S | − 2)(n − |S |), so some v ∈ S has
dS(v) ≥ |S | − 1.Maximality forces dS(v) = |S | − 1, so this gives an induced K−4 .Contradiction!
Sv
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α1(G ) + τB(G ) ≤ n2/4 for triangle-free G
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is an n-vertex triangle-free graph with m edges, then
τB(G ) ≤ m − 4m2
n2.
Corollary
If G is triangle-free, then α1(G ) + τB(G ) ≤ n2/4.
Proof.
The lemma yields
α1(G ) + τB(G ) ≤ 2m − 4m2
n2.
The upper bound is maximized when m = n2/4, yielding
α1(G ) + τB(G ) ≤ n2/4.
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α1(G ) + τB(G ) ≤ n2/4 for triangle-free G
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is an n-vertex triangle-free graph with m edges, then
τB(G ) ≤ m − 4m2
n2.
Corollary
If G is triangle-free, then α1(G ) + τB(G ) ≤ n2/4.
Proof.
The lemma yields
α1(G ) + τB(G ) ≤ 2m − 4m2
n2.
The upper bound is maximized when m = n2/4, yielding
α1(G ) + τB(G ) ≤ n2/4.
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α1(G ) + τB(G ) ≤ n2/4 for triangle-free G
Lemma (Erdos–Faudree–Pach–Spencer 1988)
If G is an n-vertex triangle-free graph with m edges, then
τB(G ) ≤ m − 4m2
n2.
Corollary
If G is triangle-free, then α1(G ) + τB(G ) ≤ n2/4.
Proof.
The lemma yields
α1(G ) + τB(G ) ≤ 2m − 4m2
n2.
The upper bound is maximized when m = n2/4, yielding
α1(G ) + τB(G ) ≤ n2/4.
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Acknowledgments
This research wasperformed while I was agraduate student at theUIUC math department.
Thank you for coming!
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Acknowledgments
This research wasperformed while I was agraduate student at theUIUC math department.
Thank you for coming!
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FIN