precoloring extension for planar graphs · distance between c 1 and c 2 is at most 4 then g is one...

Precoloring extension for planargraphs

Zdenek Dvorak Bernard Lidicky

Charles University in PragueIowa State University

AMS Sectional Meeting #1112Oct 3, 2015

Theorem (The Four Color Theorem)

Every planar graph can be colored by 4 colors.

Are 4 colors enough for planar plus one edge?

uv uv

Is it possible to precolor 2 vertices?

uv · · ·

2

Theorem (Albertson)

Let G be a planar graph and S ⊂ V (G ). If ∀u, v ∈ Sdist(u, v) ≥ 4 then any precoloring of S extends to a 5-coloring ofG .

Theorem (Dvorak, L., Postle, Mohar)

Let G be a planar graph and S ⊂ V (G ). If ∀u, v ∈ Sdist(u, v) ≥ 50000 then any precoloring of S extends to a5-list-coloring of G .

Theorem (Postle, Thomas)

If G is planar, S ⊂ V (G ) and a precoloring of S extends to allvertices at distance Ω(log |S |) from S , then it extends to a5-coloring of G .

S

3

Theorem (Grotzsch)

Every triangle-free planar graph is 3-colorable.

Theorem (Aksenov; Jensen and Thomassen)

Precoloring of any two vertices in triangle-free planar graphextends to a 3-coloring.

Theorem (Aksenov)

Every planar graph with at most 3 triangles is 3-colorable.

Theorem (Borodin, Dvorak, Kostochka, L., Yancey)

Precise description of 4-critical planar graphs with exactly 3triangles.

4

Definition (k-critical graphs)

Graph G is a k-critical graph if G is not (k − 1)-colorablebut every H ⊂ G is (k − 1)-colorable.

3-critical 4-critical 5-critical

5


All plane 4-critical graphs with 4 triangles and no 4-faces can beobtained from the Thomas-Walls sequence

...

by replacing dashed edges by edges or by Havel’s quasiedge:

.

6


All plane 4-critical graphs with 4 triangles and no 4-faces C can beobtained from the Thomas-Walls sequence by replacing dashededges by edges or by Havel’s quasiedge.


Every 4-critical plane graph with 4-triangles can be obtained fromG ∈ C by expanding some vertices of degree 3. (called patching)

w

y

z

x

w

y

z w

y

z

→

7





1

2

3

x

1

2

3 1

2

3

→

7





1

2

3

x

2

1

3

2

1

3

2

1

3

2

1

3

→

7

Conjecture (Grunbaum 63)

Planar graphs without intersecting triangles are 3-colorable.

Disproved by Havel

Conjecture (Havel)

If G is planar and mutual distance of triangles is ≥ O(1), then G is3-colorable.

Proved by Dvorak, Kra,l, Thomas.

8

Theorem (Dvorak, Kra ,l, Thomas)

If G is planar and mutual distance of triangles is ≥ O(1), then G is3-colorable.

But triangles cannot be precolored.

1 2

3

1 2

3

1 2

3

1 2

3

1 2

3

1 2

3

Winding number in quadrangulation.

9


If G is planar of girth 5, S ⊂ V (G ) and a precoloring of S extendsto all vertices at distance Ω(|S |) from S , then it extends to a3-coloring of G .

S

Girth 5 needed:

1 2

3

1 2

3

Postle: distance 100|S | is siffucient.

10

Conjecture (Dvorak, Kra ,l, Thomas)

If G is planar of girth 4, S ⊂ V (G ) and ∀u, v ∈ Sdist(u, v) ≥ Ω(1) then any precoloring of S extends to a 3-coloringof G .

Conjecture (Dvorak, Kra ,l, Thomas)

If G is planar of girth 4, S ⊂ V (G ), S consists of a vertex v and4-cycle C , and distance of v and C is ≥ Ω(1) then any precoloringof S extends to a 3-coloring of G .

vC


Second conjecture implies the first one.

- We prove the second conjecture11

Theorem (Dvorak, L.)

If G is planar of girth 4, S ⊂ V (G ), S consists of a vertex v and4-cycle C and distance of v and C is ≥ Ω(1) then any precoloringof S extends to a 3-coloring of G .

vC

Characterize when a precoloring of two 4-cycles extend.

12

Definitions (S-critical graphs)graph G = (V ,E ), S ⊂ G

G is S-critical graph for 3-coloring if for every S ⊂ H ⊂ G exists a3-coloring ϕ of S such that

• ϕ extends to a 3-coloring of H

• ϕ does not extend to a 3-coloring of G .

Note that ∅-critical graph is 4-critical.

14


Let G be planar graph with two 4-faces C1 and C2 in distance≥ Ω(1). If all triangles in G are disjoint, non-cotractible, and G is(C1,C2)-critical,

• G is obtained from framed patched Thomas-Walls, or

. . .

• G is near 3, 3-quadrangulation.

1

2

3

1

2

3

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Proof sketch

Main tool - collapse 4-faces

→

Few separating ≤ 4-cycles

C2 C1

Many separating ≤ 4-cycles

C2 C1

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Proof sketch - creating separating ≤ 4-cycles

LemmaIf C1 and C2 are in distance d , then it is possible to createseparating 4-cycle and decrease d by one.

R

C1

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Proof sketch - many separating ≤ 4-cycles

C1 C2

• collapse 4-faces without destroying separating ≤ 4-cycles

C1 C2

• describe basic graphs between separating ≤ 4-cycles• with 4-faces (21 graphs)• without 4-faces (94 graphs)

• gluing of at least ≥ 1056 (or ≥ 40 with computer) basic graphs• extends any precoloring of outer cycles, or• is Thomas-Walls, or• is almost 3, 3-quadrangulation.

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Exeptions without 4-facesExceptional graphs

Z1

Z2

O1

O2

O3

O4

T1

T2

T3

T4

T5

T6

T7

T8

RZ

4Z

5Z

6

O5

O6

O7

D1

D2

D3

D4

D5

D6

D7

D8

D9

D10

D11

A1

A2

A3

A4

A5

A5’

A6

A7

A8

A9

A10

A12

A12’

A13

X1

X2

X3

X4

X5

X5’

X6

Z4D

2

Z4D

4Z

4D

8Z

4D

9O

6D

4O

6D

9Z

4X

1b

Z4X

3Z

4X

4a

Z4X

4b

Z4X

5Z

4X

5’

Z4X

6a

Z4X

6b

Z5X

5Z

5X

5’

Z4X

2Z

6X

5Z

6X

5’

Z1A

1Z

4A

1Z

4X

1a

Z3A

8a

Z3A

8b

Z3A

9a

Z3A

9b

O4A

8O

4A

9

X5D

2X

’ 5D

2X

5D

4X

5’D

4Z

4X

2Z

4a

Z4X

2Z

4b

Z4Z

4A

1Z

4Z

4X

1Z

3A

8Z

3

Z3A

9Z

3Z

4Z

4Z

3Z

4Z

4a

Z4Z

4Z

3Z

4Z

4b

Z. Dvorák Structure of 4-critical graphs19

Lemma (Dvorak, L.)

Let G be a plane graph and C1 and C2 faces of G . If G is(C1 ∪ C2)-critical and C1 and C2 are the only ≤ 4 cycles and thedistance between C1 and C2 is at most 4 then G is one of 22graphs.

C1

C2 F

Results in a planar F -critical graph of girth 5 (outer face F ).


There are 7969 F -critical planar graphs of girth 5 with outer faceF of size 16.

20

Exeptions with 4-faces

EV1 EV2 T1 T′1 T2 T′

2 S1

S2 Q1 Q2 Q3 Q4 Q5 X1

X2 X3 X4 X5 X6 X7 J1

I4

21

Chains of small graphs

T1 Q1 T1 O6 Q1

1 3

2

1 3

12

T1 Q1 T1 EV1 S1

1 3

2

3 1

21

22

More consequences

Corollary (Dvorak, L.)

If G is an n-vertex triangle-free planar graphs with maximum

degree ∆ then G has at least(

31/∆D)n

distinct colorings, where

D is constant.

Pick n/∆D vertices S of G in mutual distance ≥ D.All 3|S| precolorings of S extend to different 3-colorings of G .

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More consequences

Let G be a graph with a face C and t triangles. (3 coloring)

• if |C | = 4 and t ≤ 1, any precoloring of C extends

• if |C | = 5 and t ≤ 1, the only C -critical graph is1

2 1

2

3 3

• if |C | = 6 and t = 0, the only C -critical graph is1

2 1

2

3 3

• if |C | ≤ 9 and t = 0, C -critical graphs known

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Let G be a graph with a 4-face C and 2 triangles. If G is C -criticalthe G is

1

2 1

2

3 3

or framed patched Thomas-Walls, where the dashed edge is anormal edge or Havel’s quaziedge.

. . .

(C is the outer face, vertices of degree 2 have different colors)

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Thank you for your attention!

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precoloring extension for planar graphs · distance between c 1 and c 2 is at most 4 then g is one...

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