Proof of a conjectured lower bound on the chromatic number of a graph

Andô, Tsuyoshi; Lin, Minghua

doi:10.1016/j.laa.2015.08.007

Cited by 28 publications

(41 citation statements)

References 3 publications

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“…Proof. Ando and Lin [1] proved a conjecture due to Wocjan and Elphick [22] that 1+s + /s − ≤ χ and consequently s + ≤ 2m(χ−1)/χ. It is clear that ψ(ψ −1) ≤ 2m.…”

Section: Definitionsmentioning

confidence: 95%

Upper bounds for the achromatic and coloring numbers of a graph

Elphick²

2017

Discrete Applied Mathematics

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Dvořák et al. introduced a variant of the Randić index of a graph and d(u) denotes the degree of a vertex u in G. The coloring number col(G) of a graph G is the smallest number k for which there exists a linear ordering of the vertices of G such that each vertex is preceded by fewer than k of its neighbors. It is well-known that χ(G) ≤ col(G) for any graph G, where χ(G) denotes the chromatic number of G. In this note, we show that for any graph G without isolated vertices, col(G) ≤ 2R ′ (G), with equality if and only if G is obtained from identifying the center of a star with a vertex of a complete graph. This extends some known results. In addition, we present some new spectral bounds for the coloring and achromatic numbers of a graph.

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“…Proof. Ando and Lin [1] proved a conjecture due to Wocjan and Elphick [22] that 1+s + /s − ≤ χ and consequently s + ≤ 2m(χ−1)/χ. It is clear that ψ(ψ −1) ≤ 2m.…”

Section: Definitionsmentioning

confidence: 95%

Upper bounds for the achromatic and coloring numbers of a graph

Elphick²

2017

Discrete Applied Mathematics

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“…When equality holds we call the coloring a Hoffman coloring. Recently, there has been some studies on finding reasonable lower bounds of χ(G) and on extending Hoffman's bound [2,7,19].…”

Section: Chromatic Number and Weight-regularitymentioning

confidence: 99%

A characterization and an application of weight-regular partitions of graphs

Abiad

2019

Linear Algebra and its Applications

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A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex u ∈ V a weight that equals the corresponding entry ν u of the Perron eigenvector ν. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity by using a Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular graphs. In addition, we show an application of weight-regular partitions to study graphs that attain equality in the classical Hoffman's lower bound for the chromatic number of a graph, and we show that weight-regularity provides a condition under which Hoffman's bound can be improved.

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“…Weakly perfect graphs have ω(G) = χ(G). Therefore using the result due to Ando and Lin [1] discussed above and that µ ≥ 2m/n:…”

Section: Proof For Weakly Perfect Graphsmentioning

confidence: 99%

“…2 Replacing µ 2 with s + Edwards and Elphick [5] proved that 2m 2m − µ 2 ≤ χ(G) and Ando and Lin [1] proved a conjecture due to Wocjan and Elphick [15] that…”

Section: Introductionmentioning

confidence: 99%