The Four Color Theorem The Four Color Theorem (Appel and Haken - - PowerPoint PPT Presentation

Coloring graphs without subdivisions of K5

Xingxing Yu

Orlando, Florida

May 18, 2019

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The Four Color Theorem

◮ The Four Color Theorem (Appel and Haken 1977): If G is a

planar graph then χ(G) ≤ 4.

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The Four Color Theorem

◮ The Four Color Theorem (Appel and Haken 1977): If G is a

planar graph then χ(G) ≤ 4.

◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a

simpler and cleaner proof.

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The Four Color Theorem

◮ The Four Color Theorem (Appel and Haken 1977): If G is a

planar graph then χ(G) ≤ 4.

◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a

simpler and cleaner proof.

◮ Theorem (Ringel and Youngs 1968): If G is embeddable in a

surface other than the sphere with Euler characteristic χ∗ then χ(G) ≤ 7 + √49 − 24χ∗ 2

• .

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The Four Color Theorem

◮ The Four Color Theorem (Appel and Haken 1977): If G is a

planar graph then χ(G) ≤ 4.

◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a

simpler and cleaner proof.

◮ Theorem (Ringel and Youngs 1968): If G is embeddable in a

surface other than the sphere with Euler characteristic χ∗ then χ(G) ≤ 7 + √49 − 24χ∗ 2

• .

◮ Conjecture (Tutte 1966): Every 2-edge-connected graph

without Petersen minor has a nowhere-zero 4-flow.

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Characterizations of planar graphs

◮ Theorem (Kuratowski 1930): A graph is planar iff it contains

no K3,3-subdivision or K5-subdivision.

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Characterizations of planar graphs

◮ Theorem (Kuratowski 1930): A graph is planar iff it contains

no K3,3-subdivision or K5-subdivision.

◮ Theorem (Wagner 1937): A graph is planar iff it contains no

K3,3-minor or K5-minor.

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Graphs containing no TK3,3

◮ Theorem (Wagner 1937): If G contains no K3,3-subdivision

then G is planar, or G ∼ = K5, or G admits a cut of size at most 2.

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Graphs containing no TK3,3

◮ Theorem (Wagner 1937): If G contains no K3,3-subdivision

then G is planar, or G ∼ = K5, or G admits a cut of size at most 2.

◮ If G does not contain K3,3-subdivision then χ(G) ≤ 5.

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Hadwiger’s conjecture

◮ Theorem (Wagner 1937): If G does not contain K5-minor and

G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K5-minor then χ(G) ≤ 4.

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Hadwiger’s conjecture

◮ Theorem (Wagner 1937): If G does not contain K5-minor and

G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K5-minor then χ(G) ≤ 4.

◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not

contain Kk+1-minor then χ(G) ≤ k.

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Hadwiger’s conjecture

◮ Theorem (Wagner 1937): If G does not contain K5-minor and

G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K5-minor then χ(G) ≤ 4.

◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not

contain Kk+1-minor then χ(G) ≤ k.

◮ Hadwiger’s conjecture is known to be true for k ≤ 5, but

remains open for k ≥ 6.

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Hadwiger’s conjecture

◮ Theorem (Wagner 1937): If G does not contain K5-minor and

G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K5-minor then χ(G) ≤ 4.

◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not

contain Kk+1-minor then χ(G) ≤ k.

◮ Hadwiger’s conjecture is known to be true for k ≤ 5, but

remains open for k ≥ 6.

◮ The case k = 5 is significantly more complex than the cases

for k ≤ 4.

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Haj´

• s conjecture

◮ Conjecture (Haj´

• s 1961): For any integer k ≥ 1, if G does not

contain Kk+1-subdivision then χ(G) ≤ k.

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Haj´

• s conjecture

◮ Conjecture (Haj´

• s 1961): For any integer k ≥ 1, if G does not

contain Kk+1-subdivision then χ(G) ≤ k.

◮ Haj´

• s’ conjecture is known to be true for k ≤ 3,

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Haj´

• s conjecture

◮ Conjecture (Haj´

• s 1961): For any integer k ≥ 1, if G does not

contain Kk+1-subdivision then χ(G) ≤ k.

◮ Haj´

• s’ conjecture is known to be true for k ≤ 3,

◮ Catlin (1979): Haj´

• s’ conjecture is false for every k ≥ 6.

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Haj´

• s conjecture

◮ Conjecture (Haj´

• s 1961): For any integer k ≥ 1, if G does not

contain Kk+1-subdivision then χ(G) ≤ k.

◮ Haj´

• s’ conjecture is known to be true for k ≤ 3,

◮ Catlin (1979): Haj´

• s’ conjecture is false for every k ≥ 6.

◮ Erd˝

• s and Fajtlowicz (1981): Haj´
• s’ conjecture is false for

almost all graphs.

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Haj´

• s conjecture

◮ Conjecture (Haj´

• s 1961): For any integer k ≥ 1, if G does not

contain Kk+1-subdivision then χ(G) ≤ k.

◮ Haj´

• s’ conjecture is known to be true for k ≤ 3,

◮ Catlin (1979): Haj´

• s’ conjecture is false for every k ≥ 6.

◮ Erd˝

• s and Fajtlowicz (1981): Haj´
• s’ conjecture is false for

almost all graphs.

◮ Thomassen (2005): Haj´

• s’ conjecture is false for many
• bvious reasons.

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Fox-Lee-Sudakov conjecture

◮ Let σ(G) denotes the largest t such that G contains

Kt-subdivision, and

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Fox-Lee-Sudakov conjecture

◮ Let σ(G) denotes the largest t such that G contains

Kt-subdivision, and

◮ Let

H(n) := max{χ(G)/σ(G) : G is a graph with |V (G)| = n}.

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Fox-Lee-Sudakov conjecture

◮ Let σ(G) denotes the largest t such that G contains

Kt-subdivision, and

◮ Let

H(n) := max{χ(G)/σ(G) : G is a graph with |V (G)| = n}.

◮ Theorem (Fox, Lee, and Sudakov, 2012):

H(n) = Θ(√n/ log n).

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Fox-Lee-Sudakov conjecture

◮ Let σ(G) denotes the largest t such that G contains

Kt-subdivision, and

◮ Let

H(n) := max{χ(G)/σ(G) : G is a graph with |V (G)| = n}.

◮ Theorem (Fox, Lee, and Sudakov, 2012):

H(n) = Θ(√n/ log n).

◮ Conjecture (Fox, Lee, and Sudakov, 2012): There is a

constant c > 0 such that every graph G with χ(G) = k satisfies σ(G) ≥ c√k log k.

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Haj´

• s conjecture

◮ Theorem (K¨

uhn and Osthus, 2002): Haj´

• s’ conjecture holds

for large k and graphs with large girth.

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Haj´

• s conjecture

◮ Theorem (K¨

uhn and Osthus, 2002): Haj´

• s’ conjecture holds

for large k and graphs with large girth.

◮ Haj´

• s’ conjecture remains open for the cases k = 4 and

k = 5.

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Haj´

• s conjecture

◮ Theorem (K¨

uhn and Osthus, 2002): Haj´

• s’ conjecture holds

for large k and graphs with large girth.

◮ Haj´

• s’ conjecture remains open for the cases k = 4 and

k = 5.

◮ Find structure of graphs containing no K5-subdivision or

K6-subdivision.

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Haj´

• s conjecture

◮ Theorem (K¨

uhn and Osthus, 2002): Haj´

• s’ conjecture holds

for large k and graphs with large girth.

◮ Haj´

• s’ conjecture remains open for the cases k = 4 and

k = 5.

◮ Find structure of graphs containing no K5-subdivision or

K6-subdivision.

◮ Find structure of a minimum counterexample to Haj´

• s’

conjecture.

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Graphs containing no TK5

◮ Theorem (He, Wang, and Y. 2017): If G does not contain

K5-subdivision then G is planar or G admits a cut of size at most 4. (This was conjectured independently by Kelmans 1979 and Seymour 1977.)

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Graphs containing no TK5

◮ Theorem (He, Wang, and Y. 2017): If G does not contain

K5-subdivision then G is planar or G admits a cut of size at most 4. (This was conjectured independently by Kelmans 1979 and Seymour 1977.)

◮ Corollary. Any counterexample to the Haj´

• s conjecture on

4-coloring must have a cut of size at most 4.

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Structure of Haj´

• s graph

A graph G is said to be a Haj´

• s graph if

(i) G is not 4-colorable, (ii) G contains no K5-subdivision, and (iii) subject to (i) and (ii), |V (G)| is minimum.

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Structure of Haj´

• s graph

◮ Theorem (Y. and Zickfeld, 2004). Every Haj´

• s graph is

4-connected. G1 G2 G′

1

G′

2

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Structure of Haj´

• s graph

◮ Theorem (Watkins and Mesner 1967): Let G be a 2-connected

graph and y1, y2, y3 be three distinct vertices in G. Then G does not have a cycle containing {y1, y2, y3} if and only if

y1 y2 y3 y1 y2 y3 y1 y2 y3 Xingxing Yu Coloring

Coloring using 4-cuts

a b c d x y z a b c d x y z a b c d x y z

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K −

4 in Haj´

• s graph

◮ Let G be a Haj´

• s graph and x1, x2, y1, y2 ∈ V (G) such that

G[{x1, x2, y1, y2}] ∼ = K −

4 and y1y2 /

∈ E(G).

y1 y2 x1 x2

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K −

4 in Haj´

• s graph

y1 y2 x1 x2

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K −

4 in Haj´

• s graph

y1 y2 x1 x2 y3

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K −

4 in Haj´

• s graph

y1 y2 x1 x2 y3

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K −

4 in Haj´

• s graph

◮ Conjecture (Lov´

asz, 1975): For every positive integer k there exists a positive integer f (k) with the following property: If G is f (k)-connected and x, y ∈ V (G) are distinct then G contains an x-y path P such that G − P is k-connected.

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K −

4 in Haj´

• s graph

◮ Conjecture (Lov´

asz, 1975): For every positive integer k there exists a positive integer f (k) with the following property: If G is f (k)-connected and x, y ∈ V (G) are distinct then G contains an x-y path P such that G − P is k-connected.

◮ True for k = 1 (Tutte), and k = 2 (Kriesell; Gould, Chen and

Y.)

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K −

4 in Haj´

• s graph

◮ Conjecture (Lov´

asz, 1975): For every positive integer k there exists a positive integer f (k) with the following property: If G is f (k)-connected and x, y ∈ V (G) are distinct then G contains an x-y path P such that G − P is k-connected.

◮ True for k = 1 (Tutte), and k = 2 (Kriesell; Gould, Chen and

Y.)

◮ Open for k ≥ 3.

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Disjoint paths

◮ G has disjoint paths from x1 to x2 and from y1 to y2.

• r

planar y1 y2 x1 x2 x1 x2 y1 y2

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Nonseparating paths

◮ G − X is a chain of blocks.

y1 y2 x1 x2 v1

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Nonseparating paths

y1 y2 x1 x2 v1 u1 u2

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Nonseparating paths

y1 y2 x1 x2 v1 v2 u1 u2

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Nonseparating paths

y1 y2 x1 x2 v1 v2 u1 u2

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Nonseparating paths

planar v1 v2 u1 u2

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Wheels in planar graphs

◮ Theorem (Xie, Xie, Y. and Yuan 2018+): Suppose G is a

Haj´

• s graph and S is a 4-cut in G. Then for each component

C of G − S with |V (C)| ≥ 2, G[C ∪ S] cannot be embedded in a closed disc with S on the boundary of the disc.

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Wheels in planar graphs

planar non-planar

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Wheels in planar graphs

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Wheels in planar graphs

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Wheels in planar graphs

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Wheels in planar graphs

x y z w u x y z w v

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Wheels in planar graphs

p q r s t u v w p q r s t

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Structure of Haj´

• s graphs with K −

4

◮ y1, y2 /

∈ X and G − X is 2-connected.

y1 y2 x1 x2

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Structure of Haj´

• s graphs with K −

4

◮ |{y1, y2} ∩ V (X)| = 1 and G − X is 2-connected. y1 x1 x2 y2

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Structure of Haj´

• s graphs with K −

4

y1 x1 x2 y2 u1 u2

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Structure of Haj´

• s graphs with K −

4

y1 x1 x2 y2

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Thank You

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