SLIDE 1
Painting Squares with 2 -1 shades Daniel W. Cranston Virginia - - PowerPoint PPT Presentation
Painting Squares with 2 -1 shades Daniel W. Cranston Virginia - - PowerPoint PPT Presentation
Painting Squares with 2 -1 shades Daniel W. Cranston Virginia Commonwealth University dcranston@vcu.edu Joint with Landon Rabern Slides available on my webpage SIAM Discrete Math 19 June 2014 Coloring Squares Coloring Squares Thm
SLIDE 2
SLIDE 3
Coloring Squares
Thm [Brooks 1941]: If ∆(G) ≥ 3 and ω(G) ≤ ∆(G) then χ(G) ≤ ∆(G).
SLIDE 4
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)
SLIDE 5
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2.
SLIDE 6
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χ(G 2) ≤ 8.
SLIDE 7
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8.
SLIDE 8
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8.
SLIDE 9
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8. Conj [C.–Kim ’08]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
SLIDE 10
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8. Conj [C.–Kim ’08]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
SLIDE 11
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8. Conj [C.–Kim ’08]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
nale
SLIDE 12
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8. Conj [C.–Kim ’08]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
nale
Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
SLIDE 13
Coloring Squares
Thm [Brooks 1941]: If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G)2. Thm [C.–Kim ’08]: If ∆(G) = 3 and ω(G 2) ≤ 8, then χℓ(G 2) ≤ 8. If G is connected and not Petersen, then ω(G 2) ≤ 8. Conj [C.–Kim ’08]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χℓ(G 2) ≤ ∆(G)2 − 1.
nale
Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G) ≥ 3, then χp(G 2) ≤ ∆(G)2 − 1.
SLIDE 14
Related Problems
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Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
SLIDE 16
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
SLIDE 17
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
SLIDE 18
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
SLIDE 19
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
Borodin–Kostochka Conjecture [1977]:
SLIDE 20
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
Borodin–Kostochka Conjecture [1977]: If ∆(G) ≥ 9 and ω(G) ≤ ∆(G) − 1, then χ(G) ≤ ∆(G) − 1.
SLIDE 21
Related Problems
Wegner’s (Very General) Conjecture [1977]: If Gk is the class of all graphs with ∆ ≤ k, then for all k ≥ 3, d ≥ 1 max
G∈Gk
χ(G d) = max
G∈Gk
ω(G d).
◮ Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.
Borodin–Kostochka Conjecture [1977]: If ∆(G) ≥ 9 and ω(G) ≤ ∆(G) − 1, then χ(G) ≤ ∆(G) − 1.
◮ Our result implies B–K conj. for G 2 when G has girth ≥ 9.
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Key Idea: d1-choosable graphs
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G).
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H.
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf:
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf:
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf:
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality.
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H).
SLIDE 30
Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H).
SLIDE 31
Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H). Its number of colors available is at least ∆2 − 1 − (dG 2(v) − dH(v))
SLIDE 32
Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H). Its number of colors available is at least ∆2 − 1 − (dG 2(v) − dH(v)) ≥ ∆2 − 1 − (∆2 − dH(v))
SLIDE 33
Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H). Its number of colors available is at least ∆2 − 1 − (dG 2(v) − dH(v)) ≥ ∆2 − 1 − (∆2 − dH(v)) = dH(v) − 1.
SLIDE 34
Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H). Its number of colors available is at least ∆2 − 1 − (dG 2(v) − dH(v)) ≥ ∆2 − 1 − (∆2 − dH(v)) = dH(v) − 1. Extend coloring to V (H), since H is d1-choosable.
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Key Idea: d1-choosable graphs
Def: A graph G is d1-choosable if it has an L-coloring whenever |L(v)| = d(v) − 1 for all v ∈ V (G). G 2 H v Lem: Minimal c/e G 2 contains no induced d1-choosable subgraph H. Pf: Color G 2 \ V (H) by minimality. Consider a vertex v ∈ V (H). Its number of colors available is at least ∆2 − 1 − (dG 2(v) − dH(v)) ≥ ∆2 − 1 − (∆2 − dH(v)) = dH(v) − 1. Extend coloring to V (H), since H is d1-choosable. Where to find d1-choosable subgraph?
SLIDE 36
Proof Outline
SLIDE 37
Proof Outline
Consider a shortest cycle C in G.
SLIDE 38
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle:
SLIDE 39
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C.
SLIDE 40
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle:
SLIDE 41
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C.
SLIDE 42
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle:
SLIDE 43
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable.
SLIDE 44
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle:
SLIDE 45
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
SLIDE 46
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
SLIDE 47
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles,
SLIDE 48
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges).
SLIDE 49
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges).
SLIDE 50
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
SLIDE 51
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
SLIDE 52
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
◮ 8+-cycle:
SLIDE 53
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
◮ 8+-cycle: similar but may
need two pendant edges.
SLIDE 54
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
◮ 8+-cycle: similar but may
need two pendant edges.
◮ 5-cycle:
SLIDE 55
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
◮ 8+-cycle: similar but may
need two pendant edges.
◮ 5-cycle: structural analysis
to find d1-choosable subgraph
SLIDE 56
Proof Outline
Consider a shortest cycle C in G.
◮ 3-cycle: dG 2(v) ≤ ∆2 − 2 for each v on C. ◮ 4-cycle: dG 2(v) ≤ ∆2 − 1 for each v on C. ◮ 6-cycle: C 2 6 is 4-regular and 3-choosable. ◮ 7-cycle: Let H be C + pendant edge.
Now since G has no shorter cycles, G 2[V (H)] ∼ = H2 (no extra edges). Use Alon–Tarsi Theorem to prove H2 is d1-choosable.
◮ 8+-cycle: similar but may
need two pendant edges.
◮ 5-cycle: structural analysis
to find d1-choosable subgraph How do we prove that (cycle + pendant edge)2 is d1-choosable?
SLIDE 57
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
SLIDE 58
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|;
SLIDE 59
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|.
SLIDE 60
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How?
SLIDE 61
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections:
SLIDE 62
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
SLIDE 63
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
1 2 3
. . .
n
SLIDE 64
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
1 2 3
. . .
n 1 2 3
. . .
n
SLIDE 65
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
1 2 3
. . .
n 1 2 3
. . .
n 1 2 3 4
. . .
n
SLIDE 66
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
1 2 3
. . .
n 1 2 3
. . .
n 1 2 3 4
. . .
n 4
. . .
n
SLIDE 67
Alon–Tarsi to prove d1-choosability
Alon–Tarsi: For a digraph D, if |EE( D)| = |EO( D)|, then D is f -choosable, where f (v) = 1 + d
D(v) for all v.
Don’t count |EE| and |EO|; just count |EE| − |EO|. How? Parity-reversing bijections: Pair most of EE and EO.
1 2 3
. . .
n 1 2 3
. . .
n 1 2 3 4
. . .
n 4
. . .
n
Lemma If Dn is the square of Cn, with all edges oriented clockwise, then |EE( Dn)| − |EO( Dn)| only depends on n (mod 3).
SLIDE 68
A Gallery of d1-choosable graphs
SLIDE 69
A Gallery of d1-choosable graphs
(a) EE=30, EO=28 (b) EE=108, EO=107 (c) EE=88, EO=87 (d) EE=512, EO=515 (e) EE=751, EO=750 (f) EE=1097, EO=1096
SLIDE 70
In Summary
SLIDE 71
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
SLIDE 72
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care?
SLIDE 73
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
SLIDE 74
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability.
SLIDE 75
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}.
SLIDE 76
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9.
SLIDE 77
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
SLIDE 78
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
◮ Where is one?
SLIDE 79
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
◮ Where is one? Shortest cycle in G + few pendant edges.
SLIDE 80
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
◮ Where is one? Shortest cycle in G + few pendant edges.
Main tool: Alon–Tarsi Theorem (for paintability)
SLIDE 81
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
◮ Where is one? Shortest cycle in G + few pendant edges.
Main tool: Alon–Tarsi Theorem (for paintability)
◮ Neat trick: Don’t count |EE| and |EO|, just |EE| − |EO|.
SLIDE 82
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.
Why do we care? Relevant to multiple conjectures.
◮ Solves conjecture of Cranston–Kim, even for paintability. ◮ Verifies Wegner’s Conjecture for d = 2 and k ∈ {4, 5}. ◮ Verifies Borodin–Kostoch Conj. for G 2 when girth(
G)≥ 9. Key idea: G 2 can’t contain induced d1-paintable subgraph.
◮ Where is one? Shortest cycle in G + few pendant edges.
Main tool: Alon–Tarsi Theorem (for paintability)
◮ Neat trick: Don’t count |EE| and |EO|, just |EE| − |EO|. ◮ How?
SLIDE 83
In Summary
Main Theorem: If G is connected and not Petersen, Hoffman–Singleton,
- r a Moore graph with ∆ = 57, then χp(G 2) ≤ ∆2 − 1.