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 [Brooks 1941] : If ∆( G ) ≥ 3 and ω ( G ) ≤ ∆( G ) then χ ( G ) ≤ ∆( G ).

Coloring Squares Thm [Brooks 1941] : If ∆( G 2 ) ≥ 3 and ω ( G 2 ) ≤ ∆( G 2 ), then χ ( G 2 ) ≤ ∆( G 2 )

Coloring Squares Thm [Brooks 1941] : If ∆( G 2 ) ≥ 3 and ω ( G 2 ) ≤ ∆( G 2 ), then χ ( G 2 ) ≤ ∆( G 2 ) ≤ ∆( G ) 2 .

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.

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 ℓ ( G 2 ) ≤ 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 χ ℓ ( G 2 ) ≤ 8. If G is connected and not Petersen, then ω ( G 2 ) ≤ 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 χ ℓ ( 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 χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1.

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 ℓ ( 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 χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1.

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 ℓ ( G 2 ) ≤ 8. If G is connected and not Petersen, then ω ( G 2 ) ≤ 8. nale Conj [C.–Kim ’08] : If G is connected, not a Moore graph, and ∆( G ) ≥ 3, then χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1.

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 ℓ ( G 2 ) ≤ 8. If G is connected and not Petersen, then ω ( G 2 ) ≤ 8. nale Conj [C.–Kim ’08] : If G is connected, not a Moore graph, and ∆( G ) ≥ 3, then χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1. Thm [C.-Rabern ’14+] : If G is connected, not a Moore graph, and ∆( G ) ≥ 3, then χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1.

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 ℓ ( G 2 ) ≤ 8. If G is connected and not Petersen, then ω ( G 2 ) ≤ 8. nale Conj [C.–Kim ’08] : If G is connected, not a Moore graph, and ∆( G ) ≥ 3, then χ ℓ ( G 2 ) ≤ ∆( G ) 2 − 1. Thm [C.-Rabern ’14+] : If G is connected, not a Moore graph, and ∆( G ) ≥ 3, then χ p ( G 2 ) ≤ ∆( G ) 2 − 1.

Related Problems

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ Our result implies Wegner’s conj. for d = 2 and k ∈ { 4 , 5 } .

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ Our result implies Wegner’s conj. for d = 2 and k ∈ { 4 , 5 } .

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ Our result implies Wegner’s conj. for d = 2 and k ∈ { 4 , 5 } .

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ Our result implies Wegner’s conj. for d = 2 and k ∈ { 4 , 5 } . Borodin–Kostochka Conjecture [1977] :

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ 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.

Related Problems Wegner’s (Very General) Conjecture [1977] : If G k is the class of all graphs with ∆ ≤ k , then for all k ≥ 3, d ≥ 1 χ ( G d ) = max ω ( G d ) . max G ∈G k G ∈G k ◮ 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.

Key Idea: d 1 -choosable graphs

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -choosable if it has an L -coloring whenever | L ( v ) | = d ( v ) − 1 for all v ∈ V ( G ).

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H .

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H . Pf:

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H . Pf:

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H . Pf:

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H . Pf: Color G 2 \ V ( H ) by minimality.

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -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 d 1 -choosable subgraph H . Pf: Color G 2 \ V ( H ) by minimality. Consider a vertex v ∈ V ( H ).

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -choosable if it has an L -coloring whenever | L ( v ) | = d ( v ) − 1 for all v ∈ V ( G ). G 2 v H Lem: Minimal c/e G 2 contains no induced d 1 -choosable subgraph H . Pf: Color G 2 \ V ( H ) by minimality. Consider a vertex v ∈ V ( H ).

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -choosable if it has an L -coloring whenever | L ( v ) | = d ( v ) − 1 for all v ∈ V ( G ). G 2 v H Lem: Minimal c/e G 2 contains no induced d 1 -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 − ( d G 2 ( v ) − d H ( v ))

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -choosable if it has an L -coloring whenever | L ( v ) | = d ( v ) − 1 for all v ∈ V ( G ). G 2 v H Lem: Minimal c/e G 2 contains no induced d 1 -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 − ( d G 2 ( v ) − d H ( v )) ≥ ∆ 2 − 1 − (∆ 2 − d H ( v ))

Key Idea: d 1 -choosable graphs Def: A graph G is d 1 -choosable if it has an L -coloring whenever | L ( v ) | = d ( v ) − 1 for all v ∈ V ( G ). G 2 v H Lem: Minimal c/e G 2 contains no induced d 1 -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 − ( d G 2 ( v ) − d H ( v )) ≥ ∆ 2 − 1 − (∆ 2 − d H ( v )) = d H ( v ) − 1.