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 West Fest A celebration of Doug’s 60th birthday! 20 June 2014

Doug’s big reveal mid-lecture.

You put 40 problems, 30 students, and a few faculty in a room; mix thoroughly, then wait for papers to precipitate out. –Doug explaining REGS

Introduction Why do we care? Coloring graphs with roughly ∆ colors Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 1 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 / 6

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 Finding big cliques: If χ = ∆, Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 6

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 Finding big cliques: If χ = ∆, then ω ≥ ⌊ ∆+1 2 ⌋ [Borodin-Kostochka ’77] Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 2 / 6

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 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 / 6

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 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 / 6

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 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 / 6

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 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 / 6

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 / 6

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 / 6

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 K 10 . Dan Cranston (VCU) Graphs with χ = ∆ have big cliques 3 / 6