List colorings of K 5 -minor-free graphs with special list - - PowerPoint PPT Presentation

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List colorings of K 5 -minor-free graphs with special list - - PowerPoint PPT Presentation

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List colorings of K 5 -minor-free graphs with special list assignments Daniel W. Cranston Virginia Commonwealth University dcranston@vcu.edu Joint with Anja Pruchnewski, Zsolt Tuza, and Margit Voigt Cycles and Colourings September 510,

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SLIDE 1

List colorings of K5-minor-free graphs with special list assignments

Daniel W. Cranston

Virginia Commonwealth University dcranston@vcu.edu

Joint with Anja Pruchnewski, Zsolt Tuza, and Margit Voigt Cycles and Colourings September 5–10, 2010

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SLIDE 2

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v).

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SLIDE 3

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v). Def: A proper L-coloring is a proper vertex coloring such that each vertex gets a color from its list L(v).

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SLIDE 4

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v). Def: A proper L-coloring is a proper vertex coloring such that each vertex gets a color from its list L(v). Def: The list-chromatic number χl(G) is the minimum k such that G has an L-coloring whenever |L(v)| ≥ k for all v ∈ V (G).

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SLIDE 5

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v). Def: A proper L-coloring is a proper vertex coloring such that each vertex gets a color from its list L(v). Def: The list-chromatic number χl(G) is the minimum k such that G has an L-coloring whenever |L(v)| ≥ k for all v ∈ V (G). We clearly have χl(G) ≥ χ(G)

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SLIDE 6

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v). Def: A proper L-coloring is a proper vertex coloring such that each vertex gets a color from its list L(v). Def: The list-chromatic number χl(G) is the minimum k such that G has an L-coloring whenever |L(v)| ≥ k for all v ∈ V (G). We clearly have χl(G) ≥ χ(G) and . . . 1,2 1,2 1,3 1,3 2,3 2,3

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SLIDE 7

List-coloring (in General)

Def: A list assignment L assigns to each v ∈ V (G) a list L(v). Def: A proper L-coloring is a proper vertex coloring such that each vertex gets a color from its list L(v). Def: The list-chromatic number χl(G) is the minimum k such that G has an L-coloring whenever |L(v)| ≥ k for all v ∈ V (G). We clearly have χl(G) ≥ χ(G) and . . . 1,2 1,2 1,3 1,3 2,3 2,3 So, χl(K3,3) > 2 = χ(K3,3).

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SLIDE 8

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable?

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SLIDE 9

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No!

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SLIDE 10

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable?

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SLIDE 11

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes

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SLIDE 12

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable.

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SLIDE 13

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χ(G) ≤ ∆(G).

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SLIDE 14

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χ(G) ≤ ∆(G).

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SLIDE 15

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G).

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SLIDE 16

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2

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SLIDE 17

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4

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SLIDE 18

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6

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SLIDE 19

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

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SLIDE 20

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
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SLIDE 21

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
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SLIDE 22

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
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SLIDE 23

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).
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SLIDE 24

List-coloring vs. coloring

Ques: Is every planar graph 4-list-colorable? No! Ques: Does ∃k s.t. every planar graph is k-list-colorable? Yes Thm 1: [Thomassen ’93] Every planar graph is 5-list-colorable. Thm 2: [Brooks ’41] If G / ∈ {Kn, C2k+1}, then χℓ(G) ≤ ∆(G). 1,2 3,4 1,2,5,6 Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

Big Question: Can we combine Theorems 1 and 3?

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SLIDE 25

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable?

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SLIDE 26

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”?

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SLIDE 27

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”?

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SLIDE 28

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

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SLIDE 29

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

◮ Need 2-connected to avoid Gallai Trees

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SLIDE 30

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

◮ Need 2-connected to avoid Gallai Trees ◮ Need 3-connected to avoid. . .

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SLIDE 31

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

◮ Need 2-connected to avoid Gallai Trees ◮ Need 3-connected to avoid. . .

0,1,2 0,1,3 0,k-2,k 0,k-1,k 1,2,. . . ,k

. . . . . .

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SLIDE 32

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

◮ Need 2-connected to avoid Gallai Trees ◮ Need 3-connected to avoid. . .

0,1,2 0,1,3 0,k-2,k 0,k-1,k 1,2,. . . ,k

. . . . . .

Why 6? (And not 5?)

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SLIDE 33

The Big Question

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Why “not complete”? Why 3-connected?

◮ Need 2-connected to avoid Gallai Trees ◮ Need 3-connected to avoid. . .

0,1,2 0,1,3 0,k-2,k 0,k-1,k 1,2,. . . ,k

. . . . . .

Why 6? (And not 5?) We have a counterexample when k = 5.

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SLIDE 34

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}.

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SLIDE 35

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk].

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SLIDE 36

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G).

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SLIDE 37

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable.

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SLIDE 38

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Proof Sketch of Main Thm:

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SLIDE 39

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Proof Sketch of Main Thm: For each component H of G[Sk], color at most 2 vertices (so that we can finish coloring H later).

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SLIDE 40

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Proof Sketch of Main Thm: For each component H of G[Sk], color at most 2 vertices (so that we can finish coloring H later). Since d(Sk) ≥ 3, each v ∈ Bk loses at most 2 colors.

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SLIDE 41

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Proof Sketch of Main Thm: For each component H of G[Sk], color at most 2 vertices (so that we can finish coloring H later). Since d(Sk) ≥ 3, each v ∈ Bk loses at most 2 colors. So |L′(v)| ≥ 5 for all v ∈ Bk. Color G[Bk] by Theorem 1’.

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SLIDE 42

Our Main Result

Def: Let Sk = {v | d(v) < k} and Bk = {v | d(v) ≥ k}. Def: Let d(Sk) be min. distance between components of G[Sk]. Main Thm: Let G be K5-minor-free, 3-connected, and not

  • complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when

f (v) = min{d(v), k} for all v ∈ V (G). Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Proof Sketch of Main Thm: For each component H of G[Sk], color at most 2 vertices (so that we can finish coloring H later). Since d(Sk) ≥ 3, each v ∈ Bk loses at most 2 colors. So |L′(v)| ≥ 5 for all v ∈ Bk. Color G[Bk] by Theorem 1’. Now finish the coloring of each H of G[Sk] (by Theorem 3).

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SLIDE 43

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

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SLIDE 44

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).
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SLIDE 45

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H

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SLIDE 46

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree.

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SLIDE 47

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2.

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SLIDE 48

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block.

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SLIDE 49

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block.

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SLIDE 50

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block.

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SLIDE 51

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block.

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SLIDE 52

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block.

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SLIDE 53

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block.

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SLIDE 54

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1

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SLIDE 55

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v)

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SLIDE 56

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v) L′(ui) = L(ui)\{a, b}

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SLIDE 57

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v) L′(ui) = L(ui)\{a, b}

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SLIDE 58

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. (5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v) L′(ui) = L(ui)\{a, b}

slide-59
SLIDE 59

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. ⇒(5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v) L′(ui) = L(ui)\{a, b}

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SLIDE 60

Main Proof

Thm 3: Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).

(adjacent (non-cut)-vertices have the same list) 5 Cases for H (0) H is not a Gallai Tree. (1) H = K1 or H = K2. (2) K2 is an end block. (3) K3 is an end block. (4) H ∈ {K3, K4} or K4 is an end block. ⇒(5) H = C2l+1 or C2l+1 is an end block. u4 u3 u2 u1 a, b ∈ L(v) L′(ui) = L(ui)\{a, b}

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SLIDE 61

(5) H = C2l+1 or C2l+1 is an end block.

v1 v2 v3 v4

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SLIDE 62

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

v1 v2 v3 v4

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SLIDE 63

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4). v1 v2 v3 v4

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SLIDE 64

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk

v1 v2 v3 v4 u0

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SLIDE 65

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk

v1 v2 v3 v4 u0 u1 u2 u3

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SLIDE 66

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk

u0 u1 u2 u3

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SLIDE 67

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk

v1 v2 v3 v4 u0 c u1 u2 u3

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SLIDE 68

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk ◮ Otherwise. . .

v1 v2 v3 v4

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SLIDE 69

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk ◮ Otherwise. . . find a K5-minor.

v1 v2 v3 v4

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SLIDE 70

(5) H = C2l+1 or C2l+1 is an end block.

◮ If ∃ vi s.t. L(vi) = L(vi+1), color vi with c ∈ L(vi) \ L(vi+1).

So assume L(v1) = . . . = L(v4).

◮ If ∃ vi s.t. N(vi) ∩ Bk = N(vi+1) ∩ Bk ◮ Otherwise. . . find a K5-minor.

v1 v2 v3 v4

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SLIDE 71

Summary

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable?

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SLIDE 72

Summary

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Main Thm: [CPTV ’10+] Let G be K5-minor-free, 3-connected, and not complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when f (v) = min{d(v), k} for all v ∈ V (G).

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SLIDE 73

Summary

Ques: [Richter] Let G be planar, 3-connected, and not complete. Let f (v) = min{d(v), 6} for all v ∈ V (G). Is G f -list-colorable? Main Thm: [CPTV ’10+] Let G be K5-minor-free, 3-connected, and not complete. If k ≥ 7 and d(Sk) ≥ 3, then G is f -list-colorable when f (v) = min{d(v), k} for all v ∈ V (G).

Tools

Thm 1’: [ˇ Skrekovski ’98] Every K5-minor-free graph is 5-list-colorable. Thm 3: [Vizing ’76, Erd˝

  • s-Rubin-Taylor ’79]

Let G be connected and let L be s.t. |L(v)| ≥ d(v) for all v ∈ V (G). If G has no L-coloring, then:

  • 1. |L(v)| = d(v) for every vertex v ∈ V (G).
  • 2. G is a Gallai tree.
  • 3. Each block B has a list L(B) and L(v) = ∪v∈BL(B).