Complexity of Fall Coloring for Restricted Graph Classes

Lauri, Juho; Mitillos, Christodoulos

doi:10.1007/s00224-020-09982-9

Cited by 3 publications

(2 citation statements)

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“…The problem of determining whether a graph is fall k-colorable (k ≥ 3) has been shown to be NP-complete [3,[11][12][13]. In this section, we give a simple proof for the NP-complete result of the FALL k-COLORABLE problem, which is defined as follows:…”

Section: Complexitymentioning

confidence: 99%

On Fall-Colorable Graphs

Wang,

Wen,

Wang

et al. 2024

Mathematics

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A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V(G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n(k−1)+r(k−r))/2, where r≡n(modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈{5,6}.

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Section: Complexitymentioning

confidence: 99%

On Fall-Colorable Graphs

Wang,

Wen,

Wang

et al. 2024

Mathematics

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“…So, we study the complexity of subfall k-coloring for k ≥ 3. By applying a result in [Lauri and Mitillos 2019], we first obtain: Theorem 1. Deciding whether a graph G has a subfall k-coloring is NP-complete for every fixed k ≥ 4.…”

Section: Complexity Of Subfall K-coloringmentioning

confidence: 99%

On the Complexity of Subfall Coloring of Graphs

Andrade¹,

Silva²

2021

Anais Do VI Encontro De Teoria Da Computação (ETC 2021)

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Given a graph G and a proper k-coloring f of G, a b-vertex in f is a vertex that is adjacent to every color class but its own. If every vertex is a b-vertex, then f is a fall k-coloring; and G has a subfall k-coloring if there is an induced H $\subseteq$ G such that H has a fall k-coloring. The subfall chromatic number of G is the largest positive integer $\psi_{fs}(G)$ such that G has a subfall $\psi_{fs}(G)$-coloring. We prove that deciding if a graph G has a subfall k-coloring is NP-complete for k $\geq$ 4, and characterize graphs having a subfall 3-coloring. This answers a question posed by Dunbar et al. (2000) in their seminal paper. We also prove polinomiality for chordal graphs and cographs, and present a comparison with other known coloring metrics based on heuristics.

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