3-coloring triangle-free planar graphs Bernard Lidick y Iowa - - PowerPoint PPT Presentation

3-coloring triangle-free planar graphs

Bernard Lidick´ y

Iowa State University O.Borodin I. Choi, J. Ekstein, Z. Dvoˇ r´ ak, P. Holub, A. Kostochka, M. Yancey.

25th Workshop on Cycles and Sep 6, 2016

Colourings

Coloring of a graph

Definition

A (proper) coloring of a graph G is a mapping ϕ : V (G) → C such that for every uv ∈ E(G) : ϕ(u) = ϕ(v). G is k-colorable if there is a (proper) coloring of G with |C| = k.

2

Coloring of a graph

Definition

A (proper) coloring of a graph G is a mapping ϕ : V (G) → C such that for every uv ∈ E(G) : ϕ(u) = ϕ(v). G is k-colorable if there is a (proper) coloring of G with |C| = k.

2

Definition of 4-critical graph

Problem: How do we efficiently describe graphs that are not 3-colorable? What are obstacles for 3-coloring? A graph G is a 4-critical graph if G is not 3-colorable but every H ⊂ G is 3-colorable. Useful as a minimal counterexample.

3

Definition of 4-critical graph

Problem: How do we efficiently describe graphs that are not 3-colorable? What are obstacles for 3-coloring? A graph G is a 4-critical graph if G is not 3-colorable but every H ⊂ G is 3-colorable. Useful as a minimal counterexample.

3

Definition of 4-critical graph

Problem: How do we efficiently describe graphs that are not 3-colorable? What are obstacles for 3-coloring? A graph G is a 4-critical graph if G is not 3-colorable but every H ⊂ G is 3-colorable. Useful as a minimal counterexample.

3

Definition of 4-critical graph

Problem: How do we efficiently describe graphs that are not 3-colorable? What are obstacles for 3-coloring? A graph G is a 4-critical graph if G is not 3-colorable but every H ⊂ G is 3-colorable. Useful as a minimal counterexample.

3

Gr¨

• tzsch’s Theorem

Theorem (Appel, Haken ’77)

Every planar graph is 4-colorable.

Theorem (Gr¨

• tzsch ’59)

Every triangle-free planar graph is 3-colorable.

4

Outline

• Proof of Gr¨
• tzsch’s Theorem
• Easy improvements
• Precolored faces
• Few triangles
• Precolored vertices

5

Proof of Gr¨

• tzsch’s Theorem - main tool

Theorem (Kostochka and Yancey ’14)

If G is a 4-critical graph, then |E(G)| ≥ 5|V (G)| − 2 3 . We write this as 3|E(G)| ≥ 5|V (G)| − 2. 4-critical graphs must have “many” edges G does not have to be planar

6

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4

Case 2: G contains no 4-faces

7

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4

Case 2: G contains no 4-faces

7

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4

Case 2: G contains no 4-faces

7

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4 x1 x2

Case 2: G contains no 4-faces

7

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4 x1 x2

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4 x1 x2

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4 x1 x2 y1 y2

Case 2: G contains no 4-faces

7

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4 x1 x2 y1 y2

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G) Case 2: G contains no 4-faces

7

v1 v2 v3 v4 x1 x2 y1 y2

Proof of Gr¨

• tzsch’s Theorem (by K-Y)

G a minimal counterexample - plane, triangle-free, 4-critical. Case 1: G contains a 4-face (try to 3-color G)

v1 v2 v3 v4 x1 x2 y1 y2

Case 2: G contains no 4-faces |E(G)| = e, |V (G)| = v, |F(G)| = f .

• v + f = e − 2 by Euler’s formula
• 5v − 10 ≥ 3e (no 3-,4-faces)
• 3e ≥ 5v − 2 (every 4-critical graph)

7

Theorem (Gr¨

• tzsch ’59)

Every planar triangle-free graph is 3-colorable. The last inequalities have a gap:

• 5v − 10 ≥ 3e (no 3-,4-faces)
• 3e ≥ 5v − 2 (every 4-critical graph)

8

Plus extra edge

Theorem (Aksenov ’77; Jensen, Thomassen ’00)

If H can be obtained from a triangle-free planar graph by adding an edge h, then H is 3-colorable.

h

9

Plus extra edge

Theorem (Aksenov ’77; Jensen, Thomassen ’00)

If H can be obtained from a triangle-free planar graph by adding an edge h, then H is 3-colorable.

h

Simple proof (Borodin, Kostochka, L., Yancey 2014). Tight K4

9

Plus extra vertex

Theorem (Jensen, Thomassen ’00)

If H can be obtained from a triangle-free planar graph by adding a vertex v of degree 3, then H is 3-colorable.

v

10

Plus extra vertex

Theorem (Borodin, Kostochka, L., Yancey ’14)

If H can be obtained from a triangle-free planar graph by adding a vertex v of degree 4, then H is 3-colorable.

v

(and the proof is simple, tight by 5-wheel)

10

Precoloring

Theorem (Gr¨

• tzsch ’59)

Let G be a triangle-free plane graph and F be the outer face of G

• f length at most 5. Then each 3-coloring of F can be extended to

a 3-coloring of G.

G G

11

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face 1 2 1 2 G Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G G H Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G G H 1 2 1 3 G H Case 2: F is a 5-face

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G G H 1 2 1 3 G H Case 2: F is a 5-face 2 3 2 1 3 G

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G G H 1 2 1 3 G H Case 2: F is a 5-face 2 3 2 1 3 G G v H

12

Simple proof

If G is a triangle-free plane graph, F is a precolored external 4-face

• r 5-face, then the precoloring of F extends.

Case 1: F is a 4-face H is 3-colorable 1 2 1 2 G G v H 1 2 1 2 G 3 H 1 2 1 3 G G H 1 2 1 3 G H Case 2: F is a 5-face 2 3 2 1 3 G G v H 2 3 2 1 3 G 1 H

12

Precolored faces

G is a plane triangle-free graph with a face bounded by a cycle C

• if |C| ≤ 5, any precoloring of C extends
• if |C| = 6, any precoloring of C extends unless G contains

1 2 1 2 3 3

by Gimbel, Thomassen ’97

13

Precolored faces

G is a plane triangle-free graph with a face bounded by a cycle C

• if |C| ≤ 5, any precoloring of C extends
• if |C| = 6, any precoloring of C extends unless G contains

1 2 1 2 3 3

by Gimbel, Thomassen ’97 Similar lists known for

• |C| = 7 Aksenov, Borodin, Glebov ’04
• |C| = 8 Dvoˇ

r´ ak, L. ’15

• |C| = 9 Choi, Ekstein, Holub, L.

13

Precolored faces

G is a plane triangle-free graph with a face bounded by a cycle C

• if |C| ≤ 5, any precoloring of C extends
• if |C| = 6, any precoloring of C extends unless G contains

1 2 1 2 3 3

by Gimbel, Thomassen ’97 Similar lists known for

• |C| = 7 Aksenov, Borodin, Glebov ’04
• |C| = 8 Dvoˇ

r´ ak, L. ’15

• |C| = 9 Choi, Ekstein, Holub, L.

Girth 5 completely solved by Dvoˇ r´ ak, Kawarabayashi ’11 Girth 4 by Dvoˇ r´ ak and Pek´ arek

13

Some triangles?

Theorem (Gr¨

• tzsch ’59)

Every planar triangle-free graph is 3-colorable. We already showed one triangle!

G

Removing one edge of triangle results in triangle-free G.

14

Some triangles?

Theorem (Gr¨

• tzsch ’59)

Every planar triangle-free graph is 3-colorable. We already showed one triangle!

G

Removing one edge of triangle results in triangle-free G.

Theorem (Gr¨

unbaum ’63; Aksenov ’74; Borodin ’97)

Let G be a planar graph containing at most three triangles. Then G is 3-colorable.

G

14

More triangles?

Theorem (Gr¨

• tzsch ’59)

Every planar triangle-free graph is 3-colorable.

Theorem (Gr¨

unbaum ’63; Aksenov ’74; Borodin ’97)

Let G be a planar graph containing at most three triangles. Then G is 3-colorable.

G

We can simplify the proof. Question: What about four triangles? Call 4-critical plane graph with four triangles a 4, 4-graph.

15

Planar graphs with four triangles?

Havel ’69 found

Problem (Sachs ’72)

Let G be a 4, 4-graph. Can the triangles be partitioned into two pairs so that in each pair the distance between the triangles is less than two?

16

Planar graphs with four triangles?

Havel ’69 found

Problem (Sachs ’72)

Let G be a 4, 4-graph. Can the triangles be partitioned into two pairs so that in each pair the distance between the triangles is less than two? No! Aksenov and Mel’nikov ’78,’80, Two infinite series of 4, 4-graphs.

16

Problem (Erd˝

• s ’90)

Describe 4, 4-graphs. Borodin ’97 - at least 15 infinite families of 4, 4-graphs (all with 4-faces) Thomas and Walls ’04 - Infinite family T W of 4, 4-graphs with no 4-faces. ...

17

Some known 4, 4-critical planar graphs

...

18

No 4-faces

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

All plane 4, 4-graphs with no 4-faces can be obtained from the Thomas-Walls sequence ... by replacing dashed edges by edges or the gadget (Havel’s quaziedge)

19

With 4-faces

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

All plane 4, 4-graphs with no 4-faces are precisely graphs in C.

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

Every 4, 4-graph can be obtained from G ∈ C by expanding some vertices of degree 3.

w y z x w y z w y z →

20

With 4-faces

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

All plane 4, 4-graphs with no 4-faces are precisely graphs in C.

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

Every 4, 4-graph can be obtained from G ∈ C by expanding some vertices of degree 3.

w y z x w y z w y z →

20

With 4-faces

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

All plane 4, 4-graphs with no 4-faces are precisely graphs in C.

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

Every 4, 4-graph can be obtained from G ∈ C by expanding some vertices of degree 3.

1 2 3 x 1 2 3 1 2 3 →

20

With 4-faces

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

All plane 4, 4-graphs with no 4-faces are precisely graphs in C.

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

Every 4, 4-graph can be obtained from G ∈ C by expanding some vertices of degree 3.

1 2 3 x 2 1 3 2 1 3 2 1 3 2 1 3 →

20

Problem (Sachs ’72)

Let G be a 4, 4-graph. Can the triangles be partitioned into two pairs so that in each pair the distance between the triangles is less than two?

Theorem (Borodin, Dvoˇ

r´ ak, Kostochka, L., Yancey ’14)

Triangles in a 4, 4-graph can be partitioned into two pairs so that in each pair the distance between the triangles is at most two.

21

Conjecture (Havel ’70)

There exists d > 0 so that if G is planar with mutual distance of triangles ≥ d, then G is 3-colorable. Proved by Dvoˇ r´ ak, Kr´ a

,

l, Thomas.

22

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar with mutual distance of triangles ≥ d, then G is 3-colorable.

23

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar with mutual distance of triangles ≥ d, then G is 3-colorable. But triangles cannot be precolored.

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Winding number in quadrangulation.

23

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 5, S ⊂ V (G) and a precoloring of S extends to all vertices at distance ≤ d|S| from S, then it extends to a 3-coloring of G.

d|S| S

24

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 5, S ⊂ V (G) and a precoloring of S extends to all vertices at distance ≤ d|S| from S, then it extends to a 3-coloring of G.

d|S| S

Girth 5 needed:

1 2 3 1 2 3 1 2 3 1 2 3

Postle: distance d = 100 is sufficient.

24

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G) and ∀u, v ∈ S dist(u, v) ≥ d, then any precoloring of S extends to a 3-coloring of G.

25

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G) and ∀u, v ∈ S dist(u, v) ≥ d, then any precoloring of S extends to a 3-coloring of G.

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G), S consists of a vertex v and 4-cycle C, and distance of v and C is ≥ d, then any precoloring of S extends to a 3-coloring of G.

v C 25

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G) and ∀u, v ∈ S dist(u, v) ≥ d, then any precoloring of S extends to a 3-coloring of G.

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G), S consists of a vertex v and 4-cycle C, and distance of v and C is ≥ d, then any precoloring of S extends to a 3-coloring of G.

v C

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

Second conjecture implies the first one.

25

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G) and ∀u, v ∈ S dist(u, v) ≥ d, then any precoloring of S extends to a 3-coloring of G.

Conjecture (Dvoˇ r´ ak, Kr´ a , l, Thomas)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G), S consists of a vertex v and 4-cycle C, and distance of v and C is ≥ d, then any precoloring of S extends to a 3-coloring of G.

v C

Theorem (Dvoˇ r´ ak, Kr´ a , l, Thomas)

Second conjecture implies the first one.

• We prove the second conjecture

25

Theorem (Dvoˇ r´ ak, L.)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G), S consists of a vertex v and 4-cycle C and distance of v and C is ≥ d, then any precoloring of S extends to a 3-coloring of G.

v C

26

Theorem (Dvoˇ r´ ak, L.)

There exists d > 0 so that if G is planar of girth 4, S ⊂ V (G), S consists of a vertex v and 4-cycle C and distance of v and C is ≥ d, then any precoloring of S extends to a 3-coloring of G.

v C

Characterize when a precoloring of two 4-cycles extend.

26

Definition - S-critical graph

For a graph G = (V , E), S ⊂ G G is an S-critical graph for 3-coloring if for every H such that S ⊂ H ⊂ G there exists a 3-coloring ϕ of S where

• 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.

27

Definition - S-critical graph

For a graph G = (V , E), S ⊂ G G is an S-critical graph for 3-coloring if for every H such that S ⊂ H ⊂ G there exists a 3-coloring ϕ of S where

• 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.

27

Definition - S-critical graph

For a graph G = (V , E), S ⊂ G G is an S-critical graph for 3-coloring if for every H such that S ⊂ H ⊂ G there exists a 3-coloring ϕ of S where

• 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.

27

Definition - S-critical graph

For a graph G = (V , E), S ⊂ G G is an S-critical graph for 3-coloring if for every H such that S ⊂ H ⊂ G there exists a 3-coloring ϕ of S where

• 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.

27

Theorem (Dvoˇ r´ ak, L.)

There exists d > 0 so that if G is a plane graph with two 4-faces C1 and C2 in distance ≥ d and all triangles in G are disjoint, non-contractible, and G is (C1, C2)-critical, then

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

. . .

• G is a near 3, 3-quadrangulation.

1 2 3 1 2 3 1 2 3 1 2 3

28

Proof sketch

Main tool - collapse 4-faces

→

Few separating ≤ 4-cycles

C2 C1

Many separating ≤ 4-cycles

C2 C1

29

Proof sketch - creating separating ≤ 4-cycles

Lemma

If C1 and C2 have distance ≥ d, then it is possible to create a separating 4-cycle and decrease d by one.

C1 C2

30

Proof sketch - creating separating ≤ 4-cycles

Lemma

If C1 and C2 have distance ≥ d, then it is possible to create a separating 4-cycle and decrease d by one.

30

C1 C2

Proof sketch - creating separating ≤ 4-cycles

Lemma

If C1 and C2 have distance ≥ d, then it is possible to create a separating 4-cycle and decrease d by one.

C1 C2

30

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 an almost 3, 3-quadrangulation.

31

Basic graphs without 4-faces

Z1 Z2 O1 O2 O3 O4 T1 T2 T3 T4 T5 T6 T7 T8 R Z4 Z5 Z6 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 Z4D2 Z4D4 Z4D8 Z4D9 O6D4 O6D9 Z4X1b Z4X3 Z4X4a Z4X4b Z4X5 Z4X5’ Z4X6a Z4X6b Z5X5 Z5X5’ Z4X2 Z6X5 Z6X5’ Z1A1 Z4A1 Z4X1a Z3A8a Z3A8b Z3A9a Z3A9b O4A8 O4A9 X5D2 X’5D2 X5D4 X5’D4 Z4X2Z4a Z4X2Z4b Z4Z4A1 Z4Z4X1 Z3A8Z3 Z3A9Z3 Z4Z4Z3Z4Z4a Z4Z4Z3Z4Z4b

32

Basic graphs 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 33

More consequences

Corollary (Dvoˇ r´ ak, L.)

If G is an n-vertex triangle-free planar graph with maximum degree ∆, then G has at least

• 31/∆Dn

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.

34

More consequences

Corollary (Dvoˇ r´ ak, L.)

If G is an n-vertex triangle-free planar graph with maximum degree ∆, then G has at least

• 31/∆Dn

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.

34

More consequences

Let G be a plane graph with a face C and t triangles.

• If |C| = 4 and t ≤ 1, then any precoloring of C extends.
• If |C| = 5 and t ≤ 1, then the only C-critical graph is:

1 2 1 2 3 3

35

Theorem (Dvoˇ r´ ak, L.)

Let G be a plane graph with a 4-face C and 2 triangles. If G is C-critical, then G is

1 2 1 2 3 3

• r a framed patched Thomas-Walls, where the dashed edge is a

normal edge or Havel’s quaziedge. . . . (C is the outer face, vertices of degree 2 have different colors)

36

Thank you for your attention!

37