Refinements of the Corrádi-Hajnal Theorem
Elyse Yeager
University of Illinois at Urbana-Champaign
Joint work with H. Kierstead and A. Kostochka
AMS Western Sectional, April 2013
Corrádi-Hajnal Theorem
Theorem (Corrádi-Hajnal 1963)
Let k ≥ 1, n ≥ 3k, and let H be an n-vertex graph with
δ(H) ≥ 2k. Then H contains k vertex-disjoint cycles.
Corrádi-Hajnal Theorem
Theorem (Corrádi-Hajnal 1963)
Let k ≥ 1, n ≥ 3k, and let H be an n-vertex graph with
δ(H) ≥ 2k. Then H contains k vertex-disjoint cycles.
This theorem is sharp.
k
k
k
Corrádi-Hajnal Theorem
Theorem (Corrádi-Hajnal 1963)
Let k ≥ 1, n ≥ 3k, and let H be an n-vertex graph with
δ(H) ≥ 2k. Then H contains k vertex-disjoint cycles.
This theorem is sharp.
k
k
k
2k − 1
Enomoto, Wang: Ore Version
Definition
σ2 (G ) = minxy 6∈E (G ) {d(x) + d(y )}
Theorem (Enomoto 1998; Wang 1999)
Let k ≥ 1, n ≥ 3k, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 1. Then H contains k vertex-disjoint cycles.
Enomoto, Wang: Ore Version
Definition
σ2 (G ) = minxy 6∈E (G ) {d(x) + d(y )}
Theorem (Enomoto 1998; Wang 1999)
Let k ≥ 1, n ≥ 3k, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 1. Then H contains k vertex-disjoint cycles.
k
k
k
2k − 1
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
k=2
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
k=2
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
k=2
k=3
Idea: G should have k vertex-disjoint cycles if:
1. |G | ≥ 3k + 1
2. α(G ) ≤ n − 2k
3. δ(G ) ≥ 2k − 1?
σ2 (G ) ≥ 4k −2?
4k −3?
k=1
k=2
k=3
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 3, n ≥ 3k + 1, and let H be an n-vertex graph with
δ(H) ≥ 2k − 1 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 3, n ≥ 3k + 1, and let H be an n-vertex graph with
δ(H) ≥ 2k − 1 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 4, n ≥ 3k + 1, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 3 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Sharpness:
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 3, n ≥ 3k + 1, and let H be an n-vertex graph with
δ(H) ≥ 2k − 1 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 4, n ≥ 3k + 1, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 3 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Sharpness:
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 3, n ≥ 3k + 1, and let H be an n-vertex graph with
δ(H) ≥ 2k − 1 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 4, n ≥ 3k + 1, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 3 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Sharpness:
2k
2k − 2
n = 4k − 2
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 3, n ≥ 3k + 1, and let H be an n-vertex graph with
δ(H) ≥ 2k − 1 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 4, n ≥ 3k + 1, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 3 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Sharpness:
2k
2k − 2
n = 4k − 2
K2t
2r
2r − 2
n = 4(r + t) − (2t + 2)
Definition
An equitable coloring of a graph G is a proper vertex coloring in
which the sizes of any two color classes differ by at most one.
Definition
An equitable coloring of a graph G is a proper vertex coloring in
which the sizes of any two color classes differ by at most one.
A graph can be equitably k-colorable, but not equitably
(k + 1)-colorable.
Definition
An equitable coloring of a graph G is a proper vertex coloring in
which the sizes of any two color classes differ by at most one.
A graph can be equitably k-colorable, but not equitably
(k + 1)-colorable.
Definition
An equitable coloring of a graph G is a proper vertex coloring in
which the sizes of any two color classes differ by at most one.
A graph can be equitably k-colorable, but not equitably
(k + 1)-colorable.
Theorem (Corrádi-Hajnal 1963)
Let H be a 3k-vertex graph with δ(H) ≥ 2k. Then H contains k
vertex-disjoint triangles.
Theorem (Corrádi-Hajnal 1963)
Let H be a 3k-vertex graph with δ(H) ≥ 2k. Then H contains k
vertex-disjoint triangles.
Corollary
Let H be a 3k-vertex graph with ∆(H) ≤ k − 1. Then H is
equitably k-colorable.
Theorem (Corrádi-Hajnal 1963)
Let H be a 3k-vertex graph with δ(H) ≥ 2k. Then H contains k
vertex-disjoint triangles.
Corollary
Let H be a 3k-vertex graph with ∆(H) ≤ k − 1. Then H is
equitably k-colorable.
Theorem (Corrádi-Hajnal 1963)
Let H be a 3k-vertex graph with δ(H) ≥ 2k. Then H contains k
vertex-disjoint triangles.
Corollary
Let H be a 3k-vertex graph with ∆(H) ≤ k − 1. Then H is
equitably k-colorable.
Theorem (Corrádi-Hajnal 1963)
Let H be a 3k-vertex graph with δ(H) ≥ 2k. Then H contains k
vertex-disjoint triangles.
Corollary
Let H be a 3k-vertex graph with ∆(H) ≤ k − 1. Then H is
equitably k-colorable.
Theorem 1 (Hajnal-Szemerédi)
If ∆(G ) ≤ k − 1, then G is equitably k-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Theorem (Kierstead, Kostochka, Y.)
For k ≥ 4, let H be a 3k + 1-vertex graph with σ2 (H) ≥ 4k − 3
and α(H) ≤ n − 2k. Then H contains k vertex-disjoint cycles.
Definition
For a graph H, let σ(H) := max{d(x) + d(y ) : xy ∈ E (H)}.
Corollary
For k ≥ 4, let H be a (3k + 1)-vertex graph with σ(H) ≤ 2k + 3
and ω(H) ≤ k + 1. Then H is equitably k + 1-colorable.
Conjecture (Chen-Lih-Wu)
If G is a (k + 1)-colorable graph with ∆(G ) ≤ k + 1, then G has an
equitable (k + 1)-coloring or k + 1 is odd and G contains Kk+1,k+1 .
Proof Outline
Theorem (Kierstead, Kostochka, Y.)
Let k ≥ 4, n ≥ 3k + 1, and let H be an n-vertex graph with
σ2 (H) ≥ 4k − 3 and α(H) ≤ n − 2k. Then H contains k
vertex-disjoint cycles.
Idea of Proof: Suppose G is an edge-maximal counterexample. Let
C be a set of disjoint cycles in G such that:
I
|C| is maximized,
I
subject to the above,
I
subject S
to both other conditions, the length of a longest path
in G − C is maximized.
P
C ∈C
|C | is minimized, and
A Small Claim
Claim
If a vertex v not in C has two neighbors in a cycle C of C, then C
has at most four vertices. If v has three neighbors to C , then C is
a triangle.
A Small Claim
Claim
If a vertex v not in C has two neighbors in a cycle C of C, then C
has at most four vertices. If v has three neighbors to C , then C is
a triangle.
A Small Claim
Claim
If a vertex v not in C has two neighbors in a cycle C of C, then C
has at most four vertices. If v has three neighbors to C , then C is
a triangle.
A Small Claim
Claim
If a vertex v not in C has two neighbors in a cycle C of C, then C
has at most four vertices. If v has three neighbors to C , then C is
a triangle.
A Small Claim
Claim
If a vertex v not in C has two neighbors in a cycle C of C, then C
has at most four vertices. If v has three neighbors to C , then C is
a triangle.
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