Claw‐Free Graphs, Skeletal Graphs, and a Stronger Conjecture on ω, Δ, and χ

a b s t r a c tCaro and Wei independently showed that the independence number α(G) of a graph G is at leastwhere ω G (u) is the maximum order of a clique of G that contains the vertex u. We discuss the relation of our conjecture to recent conjectures and results concerning the independence number and the chromatic number. Furthermore, we prove our conjecture for perfect graphs and for graphs of maximum degree at most 4.

show abstract

“…In his thesis, King [12] considered a local strengthening of Reed's conjecture. Conjecture 1.8 (King [12]).…”

Section: Conjecture 14 (Bertram and Horakmentioning

confidence: 99%

A lower bound on the independence number of a graph in terms of degrees and local clique sizes

Brause

Randerath

Rautenbach

et al. 2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Using this and the structure theory of claw-free graphs of Chudnovsky and Seymour, King [11] proved that Reed's Conjecture is true for claw-free graphs. The proof also appears in [12]. In 2013, Chudnovsky et al [4] proved that King's Conjecture holds for quasi-line graphs, and in 2015 King and Reed [12] proved it for claw-free graphs with a 3-colorable complement.…”

Section: King's Conjecturementioning

confidence: 95%

A local epsilon version of Reed's Conjecture

Kelly

Postle

2020

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

show abstract

“…A key ingredient of their proof is the result of King that every graph with

ω > \frac{2 (Δ + 1)}{3}

has a hitting set (recall that a hitting set is independent and intersects every maximum clique). About the same time, they used the Claw‐free Structure Theorem of Chudnovsky and Seymour to prove that Reed's Conjecture holds for all claw‐free graphs . Section 21.3 of Molloy and Reed gives further evidence for Reed's Conjecture by showing that the desired upper bound holds for the fractional chromatic number, even without rounding up.…”

Section: Further Directionsmentioning

confidence: 99%

Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

View full text Add to dashboard Cite

Abstract. We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon-Tarsi orientations, since analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin-Kostochka Conjecture and Reed's Conjecture.Brooks' Theorem is among the most fundamental results in graph coloring. In short, it characterizes the (very few) connected graphs for which an obvious upper bound on the chromatic number holds with equality. It has been proved and reproved using a wide range of techniques, and the different proofs generalize and extend in many directions. In this paper we share some of our favorite proofs. In addition to surveying Brooks' Theorem, we aim to illustrate many of the standard techniques in vertex coloring 1 ; furthermore, we prove versions of Brooks' Theorem for standard strengthenings of vertex coloring, including list coloring, online list coloring, and Alon-Tarsi orientations. We present the proofs roughly in order of increasing complexity, but each section is self-contained and the proofs can be read in any order. Before we state the theorem, we need a little background.A proper coloring assigns colors, denoted by positive integers, to the vertices of a graph so that endpoints of each edge get different colors. A graph G is k-colorable if it has a proper coloring with at most k colors, and its chromatic number χ(G) is the minimum value k such that G is k-colorable. If a graph G has maximum degree ∆, then χ(G) ≤ ∆ + 1, since we can repeatedly color an uncolored vertex with the smallest color not already used on its neighbors. Since the proof of this upper bound is so easy, it is natural to ask whether we can strengthen it. The answer is yes, nearly always.A clique is a subset of vertices that are pairwise adjacent. If G contains a clique K k on k vertices, then χ(G) ≥ k, since all clique vertices need distinct colors. Similarly, if G contains an odd length cycle, then χ(G) ≥ 3, even when ∆ = 2. In 1941, Brooks [10] proved that these are the only two cases in which we cannot strengthen our trivial upper bound on χ(G).

show abstract

Claw‐Free Graphs, Skeletal Graphs, and a Stronger Conjecture on ω, Δ, and χ

Cited by 25 publications

References 31 publications

A lower bound on the independence number of a graph in terms of degrees and local clique sizes

A lower bound on the independence number of a graph in terms of degrees and local clique sizes

A local epsilon version of Reed's Conjecture

Brooks' Theorem and Beyond

Contact Info

Product

Resources

About