A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

Effantin, Brice; Gastineau, Nicolas; Togni, Olivier

doi:10.1016/j.disc.2016.03.011

Cited by 12 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Effantin et al [8] introduced the relaxed b-chromatic number of a graph G, χ r b (G), as the maximum b-chomatic number of any induced subgraph of G, i.e. χ r b (G) .…”

Section: Resultsmentioning

confidence: 99%

A complexity dichotomy for critical values of the b-chromatic number of graphs

Jaffke

Lima

2020

Theoretical Computer Science

View full text Add to dashboard Cite

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χ b (G): The maximum degree ∆(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i − 1. We obtain a dichotomy result stating that for fixed k ∈ {∆(G) + 1 − p, m(G) − p}, the problem is polynomial-time solvable whenever p ∈ {0, 1} and, even when k = 3, it is NP-complete whenever p ≥ 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree ∆(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by ∆(G). Second, we show that b-Coloring is FPT parameterized by ∆(G) + k (G), where k (G) denotes the number of vertices of degree at least k.To predict the worst-case behavior of the above heuristic, Irving and Manlove defined the notions of a b-coloring and the b-chromatic number of a graph [13]. A b-coloring of a graph G is a proper coloring such that in every color class there is a vertex that has a neighbor in all of the remaining color classes, and the b-chromatic number of G, denoted by χ b (G), is the maximum integer k such that G admits a b-coloring with k colors. We observe that in a b-coloring with k colors, there is no color that can be suppressed to obtain a proper coloring with k − 1 colors, hence χ b (G) describes the worst-case behavior of the previously described heuristic on the graph G. We consider the following two computational problems associated with b-colorings of graphs. Input:Graph G, integer k Question:Does G admit a b-coloring with k colors? b-ColoringInput:Graph G, integer k Question:IsWe would like to point out an important distinction from the 'standard' notion of proper colorings of graphs: If a graph G has a b-coloring with k colors, then this implies that χ b (G) ≥ k. However, if χ b (G) ≥ k then we can in general not conclude that G has a b-coloring with k colors. A graph for which the latter implication holds as well is called b-continuous. This notion is mostly of structural interest, since the problem of determining if a graph is b-continous is NP-complete even if an optimal proper coloring and a b-coloring with χ b (G) colors are given [2]. Besides observing that χ b (G) ≤ ∆(G) + 1 where ∆(G) denotes the maximum degree of G, Irving and Manlove [13] defined the m-degree of G as the largest integer i such that G has i vertices of degree at least i − 1. It follows that χ b (G) ≤ m(G). Since the definition of the b-chromatic number originated in the analysis of the worst-case be...

show abstract

“…Recently, Effantin et al [8] introduced the relaxed b-chromatic number of a graph G, χ r b (G), as the maximum b-chomatic number of any induced subgraph of G, i.e. χ r b (G) .…”

Section: Resultsmentioning

confidence: 99%

A complexity dichotomy for critical values of the b-chromatic number of graphs

Jaffke

Lima

2020

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…Note that in a k-witness of Partial Grundy Coloring or b-Chromatic Core, each color class contains a supported vertex, which we call a center. As each center requests at most k − 1 supporting vertices, a minimal k-witnesses of Partial Grundy Coloring or b-Chromatic Core has size bounded by k 2 [14]. We note that a leaf of a center may very well be a center itself.…”

Section: Proof the Forward Direction Is Trivial Suppose That G[s] Has...mentioning

confidence: 99%

Grundy Coloring & friends, Half-Graphs, Bicliques

Aboulker,

Bonnet,

Kim

et al. 2020

Preprint

View full text Add to dashboard Cite

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f (k)n 2 k−1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative.The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on Kt,t-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.

show abstract

“…, C k such that for each i, the class C i contains a vertex say v such that v has a neighbor in any other class C j , j = i. Also by a color dominating vertex u of G we mean the vertex u has at least one neighbor in each color class of the color-dominating coloring of G. Denote by b(G) the maximum number of colors used in any color-dominating coloring of G. The b-chromatic number has been widely studied in graph theory [2,3,4,7,8,10,11,12,13]. For a survey on b-chromatic number, we refer to [9].…”

Section: Introductionmentioning

confidence: 99%

On Grundy and b-chromatic number of some families of graphs: a comparative study

Masih¹,

Zaker²

2020

Preprint

View full text Add to dashboard Cite

The Grundy and the b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G, denoted by Γ(G) is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number b(G) is the maximum number of colors used in any color-dominating coloring of G. Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence {G n } n≥1 with limited b-chromatic number but Γ(G n ) → ∞. We obtain families of graphs F such that for some adequate function f (.), Γ(G) ≤ f (b(G)), for each graph G from the family. This verifies a previous conjecture for these families.

show abstract

A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

Cited by 12 publications

References 19 publications

A complexity dichotomy for critical values of the b-chromatic number of graphs

A complexity dichotomy for critical values of the b-chromatic number of graphs

Grundy Coloring & friends, Half-Graphs, Bicliques

On Grundy and b-chromatic number of some families of graphs: a comparative study

Contact Info

Product

Resources

About