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Discharging and List coloring Bernard Lidick Department of Applied Math Charles University Winter school 2007 - Finse Bernard Lidick (Charles University) Discharging and List coloring FINSE 2007 1 / 20 Outline List coloring 1 From

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SLIDE 1

Discharging and List coloring

Bernard Lidický

Department of Applied Math Charles University

Winter school 2007 - Finse

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 1 / 20

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SLIDE 2

Outline

1

List coloring From Coloring to List Coloring Coloring vs. List Coloring

2

Discharging What is discharging? Example

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 2 / 20

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SLIDE 3

Graph Coloring

Definition

The coloring is assignment a color to every vertex.

Definition

The proper coloring is a coloring where adjacent vertices have different colors.

Definition

The chromatic number of graph is minimal number of colors needed by a proper coloring. Denoted by χ(G).

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 3 / 20

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SLIDE 4

Generalizing The Graph Coloring

Coloring: All vertices have same list

  • f possible colors.

List coloring: Every vertex has it’s

  • wn list of possible colors L(v).

Definition

The list coloring is assignment colors to the vertices from their own

  • lists. Formally c : v → L(v)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 4 / 20

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SLIDE 5

Generalizing The Graph Coloring

Coloring: All vertices have same list

  • f possible colors.

List coloring: Every vertex has it’s

  • wn list of possible colors L(v).

Definition

The list coloring is assignment colors to the vertices from their own

  • lists. Formally c : v → L(v)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 4 / 20

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SLIDE 6

Generalizing The Graph Coloring

Coloring: All vertices have same list

  • f possible colors.

List coloring: Every vertex has it’s

  • wn list of possible colors L(v).

Definition

The list coloring is assignment colors to the vertices from their own

  • lists. Formally c : v → L(v)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 4 / 20

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SLIDE 7

k-Choosable And Choosability

Definition

The graph is k-choosable if: Size of every color list is ≥ k → there is a proprer list coloring.

Definition

Choosability of graph G is minimal k such that G is k−choosable. Denoted by χℓ(G).

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 5 / 20

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SLIDE 8

k-Choosable And Choosability

Definition

The graph is k-choosable if: Size of every color list is ≥ k → there is a proprer list coloring.

Definition

Choosability of graph G is minimal k such that G is k−choosable. Denoted by χℓ(G).

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 5 / 20

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SLIDE 9

Rleationship Between Chromatic Number And Choosability

χ(G) ≤ χℓ(G) χ(G) ≤ ∆(G) + 1 and also χℓ(G) ≤ ∆(G) + 1 Exists graph G: χ(G) < χℓ(G)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 6 / 20

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SLIDE 10

Rleationship Between Chromatic Number And Choosability

χ(G) ≤ χℓ(G) χ(G) ≤ ∆(G) + 1 and also χℓ(G) ≤ ∆(G) + 1 Exists graph G: χ(G) < χℓ(G)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 6 / 20

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SLIDE 11

Rleationship Between Chromatic Number And Choosability

χ(G) ≤ χℓ(G) χ(G) ≤ ∆(G) + 1 and also χℓ(G) ≤ ∆(G) + 1 Exists graph G: χ(G) < χℓ(G)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 6 / 20

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SLIDE 12

Rleationship Between Chromatic Number And Choosability

χ(G) ≤ χℓ(G) χ(G) ≤ ∆(G) + 1 and also χℓ(G) ≤ ∆(G) + 1 Exists graph G: χ(G) < χℓ(G)

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 6 / 20

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SLIDE 13

What Is Know For Planar Graphs

Known Theorems: Every planar graph is 5-choosable. (all cycles) Every planar graph without triangles is 4-choosable. (no 3) Every planar bipartite graph is 3-choosable. (no 3, 5, 7, 9, 11, ...) There is a non 4-choosable planar graph without triangles.

Problem

Which planar graphs without triangles are 3-choosable?

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 7 / 20

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SLIDE 14

What Is Know For Planar Graphs

Known Theorems: Every planar graph is 5-choosable. (all cycles) Every planar graph without triangles is 4-choosable. (no 3) Every planar bipartite graph is 3-choosable. (no 3, 5, 7, 9, 11, ...) There is a non 4-choosable planar graph without triangles.

Problem

Which planar graphs without triangles are 3-choosable?

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 7 / 20

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SLIDE 15

The Idea Of Discharging

Take an imaginary planar counterexample. Remove reducible pieces while keepeing the planarity. Assign weights to vertices and faces. Move weights if needed and make all weights ≥ 0. So the reduced graph is not planar since for all planar graphs holds weigh < 0.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 8 / 20

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SLIDE 16

Degree Of Vertices And Faces

Vertex v: deg v = |{incident edges}|. Face f: deg f = |{incident edge sides}|. 2|E| =

  • deg v

2|E| =

  • deg f

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 9 / 20

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SLIDE 17

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 18

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 19

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 20

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 21

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 22

How To Get The Weights

Start from Euler formula for connected graph: |E| = |V| + |F| − 2 2 ∗ 2|E| + 2|E| = 6|V| + 6|F| − 12

  • 2 deg v +
  • deg f = 6|V| + 6|F| − 12
  • (2 deg v − 6) +
  • (deg f − 6) = −12

Definition

Weights w(v) = (2 deg v − 6), w(f) = (deg f − 6)

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 10 / 20

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SLIDE 23

Discharging Application

Theorem (1)

Every planar graph without triangles, 4-cycles and 5-cycles is 3-choosable.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 11 / 20

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SLIDE 24

The Reduction Part

Removing things without any effect for 3-choosability. Remove vertices of degree 1. Remove vertices of degree 2. We end with a planar graph without triangles, 4-cycles and 5-cycles and minimal vertex degree is 3.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 12 / 20

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SLIDE 25

The Reduction Part

Removing things without any effect for 3-choosability. Remove vertices of degree 1. Remove vertices of degree 2. We end with a planar graph without triangles, 4-cycles and 5-cycles and minimal vertex degree is 3.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 12 / 20

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SLIDE 26

Counting Weights

deg(?) w(v) w(f) 1

  • 4

2

  • 2

3

  • 3

4 2

  • 2

5 4

  • 1

6 6 deg(v) ≥ 3 → w(v) ≥ 0 deg(f) ≥ 6 → w(f) ≥ 0 All weights are non-negative.

  • w(v) +
  • w(f) ≥ 0

But for planar graph must hold

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 13 / 20

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SLIDE 27

Counting Weights

deg(?) w(v) w(f) 1

  • 4

2

  • 2

3

  • 3

4 2

  • 2

5 4

  • 1

6 6 deg(v) ≥ 3 → w(v) ≥ 0 deg(f) ≥ 6 → w(f) ≥ 0 All weights are non-negative.

  • w(v) +
  • w(f) ≥ 0

But for planar graph must hold

  • w(v) +
  • w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 13 / 20

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SLIDE 28

Discharging Application

Theorem (2)

Every planar graph without triangles, 5-cycles and adjacent 4-cycles is 3-choosable.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 14 / 20

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SLIDE 29

The Reduction Part

Removing things without any effect for 3-choosability. Remove vertices of degree 1. Remove vertices of degree 2. Remove 4-cycles with all vertices of degree 3. We end with a planar graph without triangles, and 5-cycles, every 4-cycle has vertex v : deg(v) ≥ 4 and minimal vertex degree is 3.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 15 / 20

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SLIDE 30

The Reduction Part

Removing things without any effect for 3-choosability. Remove vertices of degree 1. Remove vertices of degree 2. Remove 4-cycles with all vertices of degree 3. We end with a planar graph without triangles, and 5-cycles, every 4-cycle has vertex v : deg(v) ≥ 4 and minimal vertex degree is 3.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 15 / 20

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SLIDE 31

Counting Weights

deg(?) w(v) w(f) 1

  • 4

2

  • 2

3

  • 3

4 2

  • 2

5 4

  • 1

6 6 deg(v) ≥ 3 → w(v) ≥ 0 deg(v) ≥ 4 → w(v) ≥ 2 deg(f) ≥ 6 → w(f) ≥ 0 We have problems with 4-faces. The weight is -2 .

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 16 / 20

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SLIDE 32

Counting Weights

deg(?) w(v) w(f) 1

  • 4

2

  • 2

3

  • 3

4 2

  • 2

5 4

  • 1

6 6 deg(v) ≥ 3 → w(v) ≥ 0 deg(v) ≥ 4 → w(v) ≥ 2 deg(f) ≥ 6 → w(f) ≥ 0 We have problems with 4-faces. The weight is -2 .

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 16 / 20

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SLIDE 33

Dealing with 4-faces

Removing things without any effect for 3-choosability. Every 4-face f has it’s own vertex v with deg(v) ≥ 4 and w(v) ≥ 2. Reassign weights: w′(v) = w(v) − 2 w′(f) = w(f) + 2. So w′(f) ≥ 0 and w′(v) ≥ 0 and sum of all weights is same. For our graph holds w(v) + w(f) ≥ 0 For planar graph must hold w(v) + w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 17 / 20

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SLIDE 34

Dealing with 4-faces

Removing things without any effect for 3-choosability. Every 4-face f has it’s own vertex v with deg(v) ≥ 4 and w(v) ≥ 2. Reassign weights: w′(v) = w(v) − 2 w′(f) = w(f) + 2. So w′(f) ≥ 0 and w′(v) ≥ 0 and sum of all weights is same. For our graph holds w(v) + w(f) ≥ 0 For planar graph must hold w(v) + w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 17 / 20

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SLIDE 35

Dealing with 4-faces

Removing things without any effect for 3-choosability. Every 4-face f has it’s own vertex v with deg(v) ≥ 4 and w(v) ≥ 2. Reassign weights: w′(v) = w(v) − 2 w′(f) = w(f) + 2. So w′(f) ≥ 0 and w′(v) ≥ 0 and sum of all weights is same. For our graph holds w(v) + w(f) ≥ 0 For planar graph must hold w(v) + w(f) = −12

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 17 / 20

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SLIDE 36

Summary

We introduced the list coloring as a generalization of graph coloring. We described basics of the discharging method. We proved an example from list coloring.

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 18 / 20

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SLIDE 37

Open Problems

Problem

Is there a non 3-choosable graph without triangles and 5 cycles?

Problem

What if we allow 4 cycles to share a vertex but not edge?

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 19 / 20

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SLIDE 38

The End

Questions?

Bernard Lidický (Charles University) Discharging and List coloring FINSE 2007 20 / 20