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

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

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(G 2) ≤ ∆(G )2 − 1.

Coloring Squares

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

If ∆(G 2) ≥ 3 and ω(G 2) ≤ ∆(G 2), then χ(G 2) ≤ ∆(G 2)≤ ∆(G )2. 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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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. 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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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.

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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χ`(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`(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`(G

2) ≤ ∆(G )2 − 1.

The Finale

So for k = 7 our desired Moore graph exists and is unique!

Thm [C.-Rabern ’14+]: If G is connected, not a Moore graph, and ∆(G ) ≥ 3, then χp(G 2) ≤ ∆(G )2 − 1.

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(Gd) = max G∈Gk

ω(Gd).

I 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.

I Our result implies B–K conj. for G 2 when G has girth ≥ 9.

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(Gd) = max G∈Gk

ω(Gd).

I 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.

I Our result implies B–K conj. for G 2 when G has girth ≥ 9.

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(Gd) = max G∈Gk

ω(Gd).

I 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.

I Our result implies B–K conj. for G 2 when G has girth ≥ 9.

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(Gd) = max G∈Gk

ω(Gd).

I 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.

I Our result implies B–K conj. for G 2 when G has girth ≥ 9.

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(Gd) = max G∈Gk

ω(Gd).

I Our result implies Wegner’s conj. for d = 2 and k ∈ {4, 5}.

Borodin–Ko