Inequalities for the first‐fit chromatic number

Füredi, Zoltán; Gyárfás, András; Sárközy, Gábor N.; Selkow, Stanley M.

doi:10.1002/jgt.20327

Cited by 13 publications

(9 citation statements)

References 6 publications

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“…Section 3 is devoted to the concluding remarks. In particular, using a result of Füredi et al (2008), we observe that for t = 2 the above result is best possible. Next, a problem concerning the maximum First-Fit chromatic number among all graphs on n vertices which contain no cycle on 2t vertices is presented.…”

Section: Introductionmentioning

confidence: 76%

“…Using a result from Füredi et al (2008) we note that the upper bound of Theorem 2 is the best possible for t = 2. For any prime number p and integer α, Füredi et al (2008) have constructed a C 4 -free bipartite graph of Grundy number k = p α on at most 2k 2 vertices.…”

Section: Now Using the Latter Bound And (4)mentioning

confidence: 96%

“…For any prime number p and integer α, Füredi et al (2008) have constructed a C 4 -free bipartite graph of Grundy number k = p α on at most 2k 2 vertices. This shows that the upper bound in Theorem 2 can not be improved (up to a constant factor) in the case of 4-cycles.…”

Section: Now Using the Latter Bound And (4)mentioning

confidence: 99%

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First-Fit colorings of graphs with no cycles of a prescribed even length

Zaker

Soltani

2015

J Comb Optim

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The First-Fit (or Grundy) chromatic number of a graph G denoted by χ FF (G), is the maximum number of colors used by the First-Fit (greedy) coloring algorithm when applied to G. In this paper we first show that any graph G contains a bipartite subgraph of Grundy number χ FF (G)/2 + 1. Using this result we prove that for every t ≥ 2 there exists a real number c > 0 such that in every graph G on n vertices and without cycles of length 2t, any First-Fit coloring of G uses at most cn 1/t colors. It is noted that for t = 2 this bound is the best possible. A compactness conjecture is also proposed concerning the First-Fit chromatic number involving the even girth of graphs.

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Section: Introductionmentioning

confidence: 76%

Section: Now Using the Latter Bound And (4)mentioning

confidence: 96%

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First-Fit colorings of graphs with no cycles of a prescribed even length

Zaker

Soltani

2015

J Comb Optim

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“…This result follows from the existence of a finite list of graphs, called t-atoms, such that any graph with Grundy number at least t contains a t-atom as an induced subgraph. It has been proven that there exists a Nordhaus-Gaddum type inequality for the Grundy number [8,15], that there exist upper bounds for d-degenerate, planar and outerplanar graphs [2,5], and that there exist connections between the products of graphs and the Grundy number [6,1,4]. Recently, Havet and Sampaio [9] have proven that the problem of deciding if for a given graph G we have Γ(G) = ∆(G)+1, even if G is bipartite, is NP-complete.…”

Section: Introductionmentioning

confidence: 99%

On the family ofr-regular graphs with Grundy numberr+1

Gastineau¹,

Kheddouci²,

Togni³

2014

Discrete Mathematics

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The Grundy number of a graph G, denoted by Γ(G), is the largest k such that there exists a partition of V (G), into k independent sets V1, . . . , V k and every vertex of Vi is adjacent to at least one vertex in Vj , for every j < i. The objects which are studied in this article are families of r-regular graphs such that Γ(G) = r + 1. Using the notion of independent module, a characterization of this family is given for r = 3. Moreover, we determine classes of graphs in this family, in particular the class of r-regular graphs without induced C4, for r ≤ 4. Furthermore, our propositions imply results on partial Grundy number.

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“…see [1,11,12,14,19,27] for some details. Initiated by Gyárfás and Lehel, the research of many authors was focused on finding these classes of graphs for which the first-fit chromatic number is bounded in terms of the maximum clique size.…”

mentioning

confidence: 99%

First-Fit Coloring of Incomparability Graphs

Bosek¹,

Krawczyk²,

Matecki³

2013

SIAM J. Discrete Math.

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One of the simplest heuristics for obtaining a proper coloring of a graph is the first-fit algorithm. First-fit visits each vertex of the graph in the specified order and assigns to every point the least possible number. Let G be a class of incomparability graphs with bounded maximum clique size, closed under taking induced subgraphs. We prove that first-fit uses a bounded number of colors on the graphs in G iff there is an incomparability graph of clique size 2 not contained in G.

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Inequalities for the first‐fit chromatic number

Cited by 13 publications

References 6 publications

First-Fit colorings of graphs with no cycles of a prescribed even length

First-Fit colorings of graphs with no cycles of a prescribed even length

On the family ofr-regular graphs with Grundy numberr+1

First-Fit Coloring of Incomparability Graphs

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