On hitting all maximum cliques with an independent set

Rabern, Landon

doi:10.1002/jgt.20487

Cited by 20 publications

(21 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This theorem is a refinement of a result of Rabern , who proved the result when

ω \geq \frac{3}{4} (Δ + 1)

. The refinement relies on a strengthening of Haxell's Theorem ; this strengthening was implicit in Haxell's work and also in work of Aharoni, Berger, and Ziv .…”

Section: Introductionmentioning

confidence: 74%

“…Following and , we approach Theorem by characterizing the structure of the clique graph . Given a graph G and a collection

scriptC

of maximum cliques in G , we define the clique graph, denoted by

G (C)

, as follows.…”

Section: The Clique Graphmentioning

confidence: 99%

“…Kostochka's lemma (which appears in English in and ) actually tells us that if

ω > \frac{2}{3} (Δ + 1)

, no such set

scriptC

can exist. So it suffices to deal with the case

ω = \frac{2}{3} (Δ + 1)

.…”

Section: The Clique Graphmentioning

confidence: 99%

See 2 more Smart Citations

A Note on Hitting Maximum and Maximal Cliques With a Stable Set

Christofides

Edwards

King

2012

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…This theorem is a refinement of a result of Rabern , who proved the result when

ω \geq \frac{3}{4} (Δ + 1)

. The refinement relies on a strengthening of Haxell's Theorem ; this strengthening was implicit in Haxell's work and also in work of Aharoni, Berger, and Ziv .…”

Section: Introductionmentioning

confidence: 74%

“…Following and , we approach Theorem by characterizing the structure of the clique graph . Given a graph G and a collection

scriptC

of maximum cliques in G , we define the clique graph, denoted by

G (C)

, as follows.…”

Section: The Clique Graphmentioning

confidence: 99%

See 1 more Smart Citation

A Note on Hitting Maximum and Maximal Cliques With a Stable Set

Christofides

Edwards

King

2012

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Kostochka [20] proved that every graph with ω ≥ ∆ − √ ∆ + 3 2 has a hitting set. Rabern [27] extended this result to the case ω ≥ 3 4 (∆ + 1), and King [17] strengthened his argument to prove that G has a hitting set if ω > 2 3 (∆ + 1). This condition is optimal, as illustrated by the lexicographic product of an odd cycle and a clique.…”

Section: The First Main Theoremmentioning

confidence: 90%

Graphs with $\chi=\Delta$ Have Big Cliques

Cranston

Rabern²

2015

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A hitting set is an independent set that intersects every maximum clique. The reduction to the cubic case in the previous proof is an immediate consequence of more general lemmas on the existence of hitting sets [, , , ]. Schmerl extended Brooks' Theorem to all locally finite graphs, by constructing a recursive hitting set .…”

Section: Reducing To the Cubic Casementioning

confidence: 99%

Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

Self Cite

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

On hitting all maximum cliques with an independent set

Abstract: Abstract:We prove that every graph G for which (G)

Cited by 20 publications

References 7 publications

A Note on Hitting Maximum and Maximal Cliques With a Stable Set

A Note on Hitting Maximum and Maximal Cliques With a Stable Set

Graphs with $\chi=\Delta$ Have Big Cliques

Brooks' Theorem and Beyond

Contact Info

Product

Resources

About