On the separation number of a graph

Miller, Zevi; Pritikin, Dan

doi:10.1002/net.3230190604

Cited by 36 publications

(31 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The differential chromatic number is known only for special graph classes, such as Hamming graphs [13], meshes [14], hypercubes [14,15], complete binary trees [16], complete m-ary trees for odd values of m [5], other special types of trees [16], and complements of interval graphs, threshold graphs and arborescent comparability graphs [17]. Upper bounds on the differential chromatic number are given by Leung et al [6] for connected graphs and by Miller and Pritikin [7] for bipartite graphs. Note that closely related is also the radio frequency assignment problem, where n transmitters have to be assigned n frequencies, so that interfering transmitters have frequencies as far apart as possible, see e.g., [18,19].…”

Section: Related Workmentioning

confidence: 99%

“…This 1-dimensional fragmented map coloring problem is nicely captured by the maximum differential coloring problem [5,6,7,8], which we slightly generalize in this paper. Given a map, define the country graph G = (V, E) in which vertices represent countries and two vertices are connected by an edge if their corresponding countries share a non-trivial geographic boundary.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

The maximum k-differential coloring problem

Bekos

Kaufmann

Kobourov

et al. 2017

Journal of Discrete Algorithms

View full text Add to dashboard Cite

Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NPcomplete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ( 2n/3 , 2n)-differential coloring problem, even in the case where the input graph is planar.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

The maximum k-differential coloring problem

Bekos

Kaufmann

Kobourov

et al. 2017

Journal of Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…Upper and lower bounds for the cyclic antibandwidth (Sykora et al 2005;Miller and Pritikin 1989) are given as 1 2…”

Section: The Problem Formalizationmentioning

confidence: 99%

“…'', can also be considered in this context. The problem is closely related to antibandwidth problem (Leung et al 1984;Miller and Pritikin 1989) which is also known as dual bandwidth or separation number problem. Exact results for the cyclic antibandwidth are proved for some standard graphs like paths, cycles, twodimensional meshes, tori, and asymptotics are obtained for hypercubes (Raspaud et al 2009).…”

Section: Introductionmentioning

confidence: 99%

A memetic algorithm for the cyclic antibandwidth maximization problem

Bansal

Srivastava

2009

Soft Comput

View full text Add to dashboard Cite

The cyclic antibandwidth problem is to embed the vertices of a graph G of n vertices on a cycle C n such that the minimum distance (measured in the cycle) of adjacent vertices is maximized. Exact results/conjectures for this problem exist in the literature for some standard graphs, such as paths, cycles, two-dimensional meshes, and tori, but no algorithm has been proposed for the general graphs in the literature reviewed by us so far. In this paper, we propose a memetic algorithm for the cyclic antibandwidth problem (MACAB) that can be applied on arbitrary graphs. An important feature of this algorithm is the use of breadth first search generated level structures of a graph to explore a variety of solutions. A novel greedy heuristic is designed which explores these level structures to label the vertices of the graph. The algorithm achieves the exact cyclic antibandwidth of all the standard graphs with known optimal values. Based on our experiments we conjecture the cyclic antibandwidth of three-dimensional meshes, hypercubes, and double stars. Experiments show that results obtained by MACAB are substantially better than those given by genetic algorithm.

show abstract

“…Some basic relations between antibandwidth and graph parameters such as minimum and maximum degrees and the chromatic number are derived by Miller and Pritikin (1989). They have also given the characterization of n-vertex balanced forests with antibandwidth = n/2 .…”

Section: Introductionmentioning

confidence: 99%

Memetic algorithm for the antibandwidth maximization problem

Bansal¹,

Srivastava²

2010

J Heuristics

View full text Add to dashboard Cite

The antibandwidth maximization problem (AMP) consists of labeling the vertices of a n-vertex graph G with distinct integers from 1 to n such that the minimum difference of labels of adjacent vertices is maximized. This problem can be formulated as a dual problem to the well known bandwidth problem. Exact results have been proved for some standard graphs like paths, cycles, 2 and 3-dimensional meshes, tori, some special trees etc., however, no algorithm has been proposed for the general graphs. In this paper, we propose a memetic algorithm for the antibandwidth maximization problem, wherein we explore various breadth first search generated level structures of a graph-an imperative feature of our algorithm. We design a new heuristic which exploits these level structures to label the vertices of the graph. The algorithm is able to achieve the exact antibandwidth for the standard graphs as mentioned. Moreover, we conjecture the antibandwidth of some 3-dimensional meshes and complement of power graphs, supported by our experimental results.

show abstract

On the separation number of a graph

Cited by 36 publications

References 4 publications

The maximum k-differential coloring problem

The maximum k-differential coloring problem

A memetic algorithm for the cyclic antibandwidth maximization problem

Memetic algorithm for the antibandwidth maximization problem

Contact Info

Product

Resources

About