Graphs with = have big cliques Daniel W. Cranston Virginia - - PowerPoint PPT Presentation

Graphs with χ = ∆ have big cliques

Daniel W. Cranston

Virginia Commonwealth University dcranston@vcu.edu Joint with Landon Rabern Slides available on my webpage Discrete Math Days of the Northeast Wesleyan University, 5 October 2013

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9?

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9?

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2 χ = ⌈15/2⌉ = 8

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2 χ = ⌈15/2⌉ = 8 Why ∆ − 1? Kt−4

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2 χ = ⌈15/2⌉ = 8 Why ∆ − 1? Kt−4 ∆ = t

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2 χ = ⌈15/2⌉ = 8 Why ∆ − 1? Kt−4 ∆ = t, ω = t − 2

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction Why do we care?

Coloring graphs with roughly ∆ colors

Prop: For all G we have χ ≤ ∆ + 1. Thm [Brooks 1941]: If ∆ ≥ 3 and ω ≤ ∆ then χ ≤ ∆. Borodin-Kostochka Conj. (B-K) [1977]: If ∆ ≥ 9 and ω ≤ ∆ − 1 then χ ≤ ∆ − 1. Why ∆ ≥ 9? ∆ = 8, ω = 6, α = 2 χ = ⌈15/2⌉ = 8 Why ∆ − 1? Kt−4 ∆ = t, ω = t − 2 χ = (t − 4) + 3 = t − 1

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

then ω ≥ ⌊ 2∆+1

3

⌋ [Mozhan ’83]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

then ω ≥ ⌊ 2∆+1

3

⌋ [Mozhan ’83] then ω ≥ ∆ − 28 [Kostochka ’80]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

then ω ≥ ⌊ 2∆+1

3

⌋ [Mozhan ’83] then ω ≥ ∆ − 28 [Kostochka ’80] then ω ≥ ∆ − 3 when ∆ ≥ 31 [Mozhan ’87]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

then ω ≥ ⌊ 2∆+1

3

⌋ [Mozhan ’83] then ω ≥ ∆ − 28 [Kostochka ’80] then ω ≥ ∆ − 3 when ∆ ≥ 31 [Mozhan ’87] then ω ≥ ∆ − 3 when ∆ ≥ 13 [C.-Rabern ’13+]

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Introduction What do we know?

Previous Results

B-K Conjecture is true for claw-free graphs [C.-Rabern ’13] B-K Conjecture is true when ∆ ≥ 1014 [Reed ’98] and likely ∆ ≥ 106 suffices B-K Conjecture is true, if it is true when χ = ∆ = 9 [Kostochka ’80] Finding big cliques: If χ = ∆,

then ω ≥ ⌊ ∆+1

2 ⌋ [Borodin-Kostochka ’77]

then ω ≥ ⌊ 2∆+1

3

⌋ [Mozhan ’83] then ω ≥ ∆ − 28 [Kostochka ’80] then ω ≥ ∆ − 3 when ∆ ≥ 31 [Mozhan ’87] then ω ≥ ∆ − 3 when ∆ ≥ 13 [C.-Rabern ’13+] then ω ≥ ∆ − 4 for all ∆

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10. Main Theorem: Every graph with χ = ∆ ≥ 13 contains K∆−3.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10. Main Theorem: Every graph with χ = ∆ ≥ 13 contains K∆−3. Proof: Let G be minimal counterexample. ∆ ≥ 14 by Lemma 2.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10. Main Theorem: Every graph with χ = ∆ ≥ 13 contains K∆−3. Proof: Let G be minimal counterexample. ∆ ≥ 14 by Lemma 2. If ω = ∆ − 4, then let I be a hitting set expanded to be a maximal independent set; otherwise let I be any maximal independent set.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10. Main Theorem: Every graph with χ = ∆ ≥ 13 contains K∆−3. Proof: Let G be minimal counterexample. ∆ ≥ 14 by Lemma 2. If ω = ∆ − 4, then let I be a hitting set expanded to be a maximal independent set; otherwise let I be any maximal independent set. If ∆(G − I) ≤ ∆(G) − 2, then win by Brooks’ Theorem.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Outline

Main Theorem

Def: A hitting set is independent set intersecting every maximum clique. Lemma 1: Every G with χ = ∆ ≥ 14 and ω = ∆ − 4 has a hitting set. Lemma 2: If G has χ = ∆ = 13, then G contains K10. Main Theorem: Every graph with χ = ∆ ≥ 13 contains K∆−3. Proof: Let G be minimal counterexample. ∆ ≥ 14 by Lemma 2. If ω = ∆ − 4, then let I be a hitting set expanded to be a maximal independent set; otherwise let I be any maximal independent set. If ∆(G − I) ≤ ∆(G) − 2, then win by Brooks’ Theorem. If ∆(G − I) = ∆(G) − 1, then G − I is a smaller counterexample, contradiction!

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I. Pr(Euv) =

1 |Su| 1 |Sv| = k−2.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I. Pr(Euv) =

1 |Su| 1 |Sv| = k−2.

Euv is independent of all but 2k(∆ − (k − 1)) = 20k events.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I. Pr(Euv) =

1 |Su| 1 |Sv| = k−2.

Euv is independent of all but 2k(∆ − (k − 1)) = 20k events. Finally, 4(20k)k−2 ≤ 1

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I. Pr(Euv) =

1 |Su| 1 |Sv| = k−2.

Euv is independent of all but 2k(∆ − (k − 1)) = 20k events. Finally, 4(20k)k−2 ≤ 1 ⇔ k ≥ 80

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Induction Step

Random Hitting Sets

Lov´ asz Local Lemma: Suppose we do a random experiment. Let E = {E1, E2, . . .} be a set of bad events such that Pr(Ei) ≤ p < 1 for all i, and each Ei is mutually independent of all but d events. If 4dp ≤ 1, then with positive probability no bad events occur. Lemma 1’: Every G with χ = ∆ ≥ 89 and ω = ∆ − 4 has a hitting set I. Proof: Get disjoint cliques S1, S2, . . . of size k := ∆ − 9 so each maximum clique contains one. To form I, choose one vertex from each Si randomly. For each edge uv with endpoints u, v in distinct Si, event Euv is that u, v both chosen for I. Pr(Euv) =

1 |Su| 1 |Sv| = k−2.

Euv is independent of all but 2k(∆ − (k − 1)) = 20k events. Finally, 4(20k)k−2 ≤ 1 ⇔ k ≥ 80 ⇔ ∆ ≥ 89.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 4 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. The club R containing v is a K4.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. The club R containing v is a K4. All other clubs are 3-colorable.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. v V1 . . . V4 The club R containing v is a K4. All other clubs are 3-colorable.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. v V1 . . . V4 The club R containing v is a K4. All other clubs are 3-colorable. For w ∈ V (R) and j ∈ {1, . . . , 4}: If dVj(w) = 3, then G[Vj + w] has a K4 component.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. v V1 . . . V4 The club R containing v is a K4. All other clubs are 3-colorable. For w ∈ V (R) and j ∈ {1, . . . , 4}: If dVj(w) = 3, then G[Vj + w] has a K4 component. For w ∈ V (R) and j ∈ {1, . . . , 4}: If w has 2 neighbors in club S of clubhouse Vi, then χ(S + w) = 4.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Setup

Clubs and Clubhouses

Def: A Mozhan Partition of a graph G with ∆ = 13 is a partition of V into clubhouses V1, . . . , V4 and a vertex v with certain properties. For each Vi, components of G[Vi] are clubs meeting in clubhouse Vi. v V1 . . . V4 The club R containing v is a K4. All other clubs are 3-colorable. For w ∈ V (R) and j ∈ {1, . . . , 4}: If dVj(w) = 3, then G[Vj + w] has a K4 component. For w ∈ V (R) and j ∈ {1, . . . , 4}: If w has 2 neighbors in club S of clubhouse Vi, then χ(S + w) = 4. Lem: Every ∆-critical graph with ∆ = 13 has a Mozhan partition.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 5 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. v w u

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. v w u

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v v x w u Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v v x w u Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v x w v u Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v x v u w Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v x v u w Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v x v u w Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10. Claim 3: Each club is active at most three times.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Results The Base Case

The Vertex Shuffle

Lemma 2: If G has χ = ∆ = 13, then G has a K10. Pf Idea: Start with a Mozhan partition of G. Repeatedly send a member

• f the active K4 to a clubhouse where it has only 3 neighbors (forming a

new K4), always at least 2 options. Move each vertex only once. Never move between clubs joined to each other. Find either a 12-coloring or K10. Claim 1: No clubs become (in)complete to each other. w u v x v u w Claim 2: If G has K4 joined to K3’s in two other clubhouses, then G has K10. Claim 3: Each club is active at most three times. Claim 4: G contains K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 6 / 8

Future Work The Iceberg

What next?

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 7 / 8

Future Work The Iceberg

What next?

The four-colour theorem is the tip of the iceberg, the thin end of the wedge, and the first cuckoo of Spring. –William Tutte

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 7 / 8

Future Work The Iceberg

What next?

The four-colour theorem is the tip of the iceberg, the thin end of the wedge, and the first cuckoo of Spring. –William Tutte Reed’s Conjecture: χ ≤ ω+∆+1

2

• .

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 7 / 8

Future Work The Iceberg

What next?

The four-colour theorem is the tip of the iceberg, the thin end of the wedge, and the first cuckoo of Spring. –William Tutte Reed’s Conjecture: χ ≤ ω+∆+1

2

• .

Theorem (Reed): There exists ǫ > 0 such that χ ≤ ⌈ǫω + (1 − ǫ)(∆ + 1)⌉.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 7 / 8

Future Work The Iceberg

What next?

The four-colour theorem is the tip of the iceberg, the thin end of the wedge, and the first cuckoo of Spring. –William Tutte Reed’s Conjecture: χ ≤ ω+∆+1

2

• .

Theorem (Reed): There exists ǫ > 0 such that χ ≤ ⌈ǫω + (1 − ǫ)(∆ + 1)⌉. Conjectured that ǫ = 1

2 works.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 7 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Local Lemma for ∆ ≥ 89.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Local Lemma for ∆ ≥ 89. Smaller ∆ are trickier, but it works for ∆ ≥ 14.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Local Lemma for ∆ ≥ 89. Smaller ∆ are trickier, but it works for ∆ ≥ 14.

Mozhan Partitions and Vertex Shuffle show that if ∆ = 13, then χ ≤ 12 or G has K10.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Local Lemma for ∆ ≥ 89. Smaller ∆ are trickier, but it works for ∆ ≥ 14.

Mozhan Partitions and Vertex Shuffle show that if ∆ = 13, then χ ≤ 12 or G has K10.

Idea: a partial coloring minimizing number of edges within clubhouses.

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8

Summary

In Review

B-K Conj: Every graph with χ = ∆ ≥ 9 contains K∆. If true, then best possible. True for claw-free graphs, and also for large ∆. Main Result: Every graph with χ = ∆ ≥ 13 contains K∆−3. Hitting sets reduce to the case ∆ = 13.

Local Lemma for ∆ ≥ 89. Smaller ∆ are trickier, but it works for ∆ ≥ 14.

Mozhan Partitions and Vertex Shuffle show that if ∆ = 13, then χ ≤ 12 or G has K10.

Idea: a partial coloring minimizing number of edges within clubhouses.

The Iceberg (Reed’s Conj): χ ≤ ω+∆+1

2

• .

Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 8 / 8