A Fractional Analogue of Brooks' Theorem

King, Andrew D.; Lü, Linyuan; Peng, Xing

doi:10.1137/110827879

Cited by 14 publications

(15 citation statements)

References 6 publications

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“…By the theorem above, we get f (∆) ≥ 1 8 for each ∆ ≥ 4. Therefore, Theorem 1 improves the result of King, Lu, and Peng [12] for each ∆ ≥ 4 and the result of Edwards and King [3] for ∆ ∈ {6, 7, 8}.…”

Section: Introductionsupporting

confidence: 71%

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The fractional chromatic number of $K_Δ$-free graphs

Peng

2021

Preprint

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Section: Introductionsupporting

confidence: 71%

“…Before Dvořák, Sereni, and Volec [2] proved Heckman and Thomas' Conjecture, progress towards this conjecture can be found in [5,8,11,13,14]. King, Lu, and Peng [12] first proved that f (∆) ≥ 2 67 for each ∆ ≥ 4,. Edwards and King [3] established an improved lower bound on f (∆) for all ∆ ≥ 6.…”

Section: Introductionmentioning

confidence: 99%

The fractional chromatic number of $K_Δ$-free graphs

Peng

2021

Preprint

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“…(A standard vertex coloring is the special case when the weight on each set is 0 or 1.) King, Lu, and Peng [34] strengthened the result of Albertson et al [1] by showing that every connected K -free graph with ≥ 4 (except for the two graphs in Figure 8) has fractional chromatic number at most − 2 67 ; this result was further strengthened by Edwards and King [23,22]. Recently Dvořák, Sereni, and Volec [21] proved fascinating results on fractional coloring of triangle-free cubic graphs.…”

Section: Kernel Perfectionmentioning

confidence: 83%

Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

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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).

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“…Moreover if ∆ ≥ 4 and G is not any of the graphs listed above, then η(G) ≤ ∆ − 2 67 . ( [15]) One question is whether we can get better bounds using the potentially denser natural sparse cuts. Equivalently, is the fractional chromatic number η V (G pack A,J ) much smaller when we consider the sparse cut support list V corresponding to the natural sparse closure?…”

Section: If G Packmentioning

confidence: 99%

Analysis of Sparse Cutting Planes for Sparse MILPs with Applications to Stochastic MILPs

Dey¹,

Molinaro²,

Wang³

2018

Mathematics of OR

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In this paper, we present an analysis of the strength of sparse cutting-planes for mixed integer linear programs (MILP) with sparse formulations. We examine three kinds of problems: packing problems, covering problems, and more general MILPs with the only assumption that the objective function is non-negative. Given a MILP instance of one of these three types, assume that we decide on the support of cutting-planes to be used and the strongest inequalities on these supports are added to the linear programming relaxation. Call the optimal objective function value of the linear programming relaxation together with these cuts as z cut . We present bounds on the ratio of z cut and the optimal objective function value of the MILP that depends only on the sparsity structure of the constraint matrix and the support of sparse cuts selected, that is, these bounds are completely data independent. These results also shed light on the strength of scenario-specific cuts for two stage stochastic MILPs.

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A Fractional Analogue of Brooks' Theorem

Abstract: Let ∆(G) be the maximum degree of a graph G. Brooks' theorem states that the only connected graphs with chromatic number χ(G) = ∆(G) + 1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper.

Cited by 14 publications

References 6 publications

The fractional chromatic number of $K_Δ$-free graphs

The fractional chromatic number of $K_Δ$-free graphs

Brooks' Theorem and Beyond

Analysis of Sparse Cutting Planes for Sparse MILPs with Applications to Stochastic MILPs

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