New bounds for the chromatic number of graphs

Zaker, Manouchehr

doi:10.1002/jgt.20298

Cited by 11 publications

(5 citation statements)

References 13 publications

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“…6. Our new bound generalizes and strengthens several bounds for proper coloring stated in terms of various potential functions (see, e.g., Borowiecki and Rautenbach 2015;Stacho 2001 andZaker 2008).…”

Section: Related Research On Proper Coloring and Our Resultsmentioning

confidence: 99%

Computational aspects of greedy partitioning of graphs

Borowiecki

2017

J Comb Optim

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In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related too greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs P-coloring with at least k colors is NPcomplete if P is a class of K p -free graphs with p ≥ 3. On the other hand we give a polynomial-time algorithm when k is fixed and the family of minimal forbidden graphs defining the class P is finite. We also prove coNP-completeness of deciding if for a given graph G and an integer t ≥ 0 the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any P-coloring of G is bounded by t. In view of computational hardness, we present new Brooks-type bound on the largest number of colors used by the greedy P-coloring algorithm.

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Section: Related Research On Proper Coloring and Our Resultsmentioning

confidence: 99%

Computational aspects of greedy partitioning of graphs

Borowiecki

2017

J Comb Optim

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“…A stronger upper bound for Grundy number in terms of vertex degree was obtained in [32] as follows. For any graph G and u ∈ V (G) we denote {v ∈ V (G) :…”

Section: Introductionmentioning

confidence: 99%

More bounds for the Grundy number of graphs

Tang

et al. 2015

J Comb Optim

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A coloring of a graph G = (V, E) is a partition {V 1 , V 2 , . . . , V k } of V into independent sets or color classes. A vertex v ∈ V i is a Grundy vertex if it is adjacent to at least one vertex in each color class V j for every j < i. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number Γ(G) of a graph G is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new characterization for a {P 4 , C 4 }-free graph by supporting a conjecture of Zaker, which says that Γ(G) ≥ δ(G) + 1 for any C 4 -free graph G.

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“…Actually, the situation is even worse: there does not exist any polynomial-time approximation scheme to estimate Γ(G) unless P = NP [20], and for every integer c it is coNP-complete to decide whether Γ(G) ≤ c χ(G), and also whether Γ(G) ≤ c ω(G), where ω(G) denotes the clique number of G (see [1]). Several bounds on Γ(G) in terms of other graph invariants were given, e.g., in [5,27,28]. On the other hand, by the finite basis theorem of Gyárfás et al [13] the problem of deciding whether Γ(G) ≥ k can be solved in polynomial time, when k is a fixed integer (see also [6] for results on Grundy critical graphs).…”

Section: Introductionmentioning

confidence: 99%

Minimum order of graphs with given coloring parameters

Bacsó

Borowiecki

Hujter

et al. 2015

Discrete Mathematics

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A complete k-coloring of a graph G = (V, E) is an assignment ϕ : V → {1, . . . , k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number χ(G) and achromatic number ψ(G), respectively), and the Grundy number Γ(G) defined as the largest k admitting a complete coloring ϕ with exactly *

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New bounds for the chromatic number of graphs

Cited by 11 publications

References 13 publications

Computational aspects of greedy partitioning of graphs

Computational aspects of greedy partitioning of graphs

More bounds for the Grundy number of graphs

Minimum order of graphs with given coloring parameters

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