On Six Problems Posed by Jarik Nešetřil

Bang‐Jensen, Jørgen; Reed, Bruce; Schacht, Mathias; Šámal, Robert; Toft, Bjarne; Wagner, Uli

doi:10.1007/3-540-33700-8_30

Cited by 15 publications

(29 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One may also wonder if the bound in Corollary 1 is tight. Theorems 3 and 4 indicate that the strong chromatic index of unit distance graphs in 2  and 3  may be small, even linear in Δ, and that is what we believe. It would also be desirable to determine exactly the maximum possible size of a strong clique in 2  and 3  .…”

Section: Casesupporting

confidence: 52%

“…Theorems 3 and 4 indicate that the strong chromatic index of unit distance graphs in 2  and 3  may be small, even linear in Δ, and that is what we believe. It would also be desirable to determine exactly the maximum possible size of a strong clique in 2  and 3  . In the plane, we can construct a strong clique of size Δ 3 − 3-take an equilateral triangle and add Δ − 2 pendant edges to each vertex.…”

Section: Casesupporting

confidence: 52%

“…Theorem 4 Let G be a unit distance graph in 2  of maximum degree Δ. The size of a strong clique in G is at most Δ 6 − 3.…”

Section: Strong Cliquesmentioning

confidence: 99%

“…The line graph of a graph G, denoted L G ( ), is a graph L such that V L E G ( ) = ( ), and for every two 2 iff e and f are joined in G. For an edge e of G, we denote by N e ( ) G the set of vertices of G that are adjacent to at least one vertex of e and not contained in e. On the other hand, N e ( )…”

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

Strong chromatic index of unit distance graphs

Dębski

2018

Journal of Graph Theory

View full text Add to dashboard Cite

The strong chromatic index of a graph G, denoted by s ′ ( G ), is defined as the least number of colors in a coloring of edges of G, such that each color class is an induced matching (or: if edges e and f have the same color, then both vertices of e are not adjacent to any vertex of f). A graph G is a unit distance graph in R n if vertices of G can be uniquely identified with points in R n, so that u v is an edge of G if and only if the Euclidean distance between the points identified with u and v is 1. We would like to find the largest possible value of s ′ ( G ), where G is a unit distance graph (in R 2 and R 3) of maximum degree Δ. We show that s ′ ( G ) ≤ K Δ 2 ln Δ, where G is a unit distance graph in R 3 of maximum degree Δ. We also show that the maximum possible size of a strong clique in unit distance graph in R 3 is linear in Δ and give a tighter result 6 Δ − 3 for unit distance graphs in the plane.

show abstract

Section: Casesupporting

confidence: 52%

Section: Casesupporting

confidence: 52%

“…Theorem 4 Let G be a unit distance graph in 2  of maximum degree Δ. The size of a strong clique in G is at most Δ 6 − 3.…”

Section: Strong Cliquesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Strong chromatic index of unit distance graphs

Dębski

2018

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Since then, the problem has proved to be elusive with no new positive or negative results. The question on the complexity and further approximability of Maximum Clique on general disk graphs is considered as folklore [8], but was also explicitly mentioned as an open problem by Fishkin [22], Ambühl and Wagner [5]. Cabello even asked if there is a 1.99-approximation for disk graphs with two sizes of radii [15,16].…”

Section: Previous Resultsmentioning

confidence: 99%

EPTAS for Max Clique on Disks and Unit Balls

Bonamy

Bonnet

Bousquet

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

We propose a polynomial-time algorithm which takes as input a finite set of points of R 3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane.Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzążewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VCdimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.

show abstract