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? Why ∆ − 1? K t − 4 ∆ = 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? Why ∆ − 1? K t − 4 ∆ = t ∆ = 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? Why ∆ − 1? K t − 4 ∆ = t , ω = t − 2 ∆ = 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? Why ∆ − 1? K t − 4 ∆ = t , ω = t − 2 ∆ = 8, ω = 6, α = 2 χ = ( t − 4) + 3 = t − 1 χ = ⌈ 15 / 2 ⌉ = 8 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 ∆ ≥ 10 14 [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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ⌋ [Mozhan ’83] 3 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ⌋ [Mozhan ’83] 3 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ⌋ [Mozhan ’83] 3 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 ∆ ≥ 10 14 [Reed ’98] and likely ∆ ≥ 10 6 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 ⌋ [Mozhan ’83] 3 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