Optimal Number Of Colors Research Articles

Optimal exclusive perpetual grid exploration by luminous myopic opaque robots with common chirality

We consider swarms of luminous myopic opaque robots that run in synchronous Look-Compute-Move cycles. These robots have no global compass, but agree on a common chirality. In this context, we propose optimal solutions to the perpetual exploration of a finite grid. Precisely, we investigate optimality in terms of visibility range, number of robots, and number of colors. In more detail, under the optimal visibility range one, we give an algorithm which is optimal w.r.t. both the number of robots and colors: it uses two robots and three colors. Under visibility two, we design two algorithms: the first one uses three robots with an optimal number of colors (i.e., one), the second one achieves the best trade-off between the number of robots and colors, i.e., it uses two robots and two colors.

Scheduling algorithms for an expatriate assignment using vertex coloring of graph

Scheduling system for an expatriate assignment needs an effective and efficient way to be designed since the expatriates reside outside their countries to do their assignments in very limited time. The problem of this scheduling system is quite complicated as the expatriates usually work in a team to do more than one assignment in some divisions. This situation leads to the frequent clash of schedules. In this paper, an algorithm for scheduling system was developed to optimize the expatriate assignment. The methods used for developing the algorithms was vertex coloring of graph. This algorithm resulted in the optimal number of colors which is equivalent to the optimal time allocated for the expatriate assignment without any clash schedule among expatriates. The algorithm can be used for designing a software or application for scheduling system.

On rainbow-k-connectivity of random graphs

A path in an edge-colored graph is called a rainbow path if the edges on it have distinct colors. For k⩾1, the rainbow-k-connectivity of a graph G, denoted by rck(G), is the minimum number of colors required to color the edges of G in such a way that every two distinct vertices are connected by at least k internally vertex-disjoint rainbow paths. In this paper, we study rainbow-k-connectivity in the setting of random graphs. We show that for every fixed integer d⩾2 and every k⩽O(logn), p=(logn)1/d/n(d−1)/d is a sharp threshold function for the property rck(G(n,p))⩽d. This substantially generalizes a result in [Y. Caro, A. Lev, Y. Roditty, Z. Tuza, R. Yuster, On rainbow connection, Electron. J. Comb. 15 (2008)], stating that p=logn/n is a sharp threshold function for the property rc1(G(n,p))⩽2. As a by-product, we obtain a polynomial-time algorithm that makes G(n,p) rainbow-k-connected using at most one more than the optimal number of colors with probability 1−o(1), for all k⩽O(logn) and p=n−ϵ(1±o(1)) for any constant ϵ∈[0,1).

Greedy Algorithms for Triangle Free Coloring

According to [2] the capability of providing good bounds for the size of a maximum clique in a graph is a critical ingredient of an efficient clique search algorithm. In order to establish upper bounds of the size of a maximum clique we introduce some new coloring schemes using s-clique free partitioning. A complexity result will show that finding the optimal number of colors is an NP-complete problem. This can be interpreted loosely as a computational limitation. To overcome this difficulty for practical purposes we suggest various greedy coloring procedures.

A Non-Conventional Coloring of the Edges of a Graph

Coloring the nodes of a graph is a commonly used technique to speed up clique search algorithms. Coloring the edges of the graph as a preconditioning method can also be used to speed up computations. In this paper we will show that an unconventional coloring scheme of the edges leads to an NP-complete problem when one intends to determine the optimal number of colors.

Approximate constrained bipartite edge coloring

We study the following Constrained Bipartite Edge Coloring problem: We are given a bipartite graph G=( U, V, E) of maximum degree l with n vertices, in which some of the edges have been legally colored with c colors. We wish to complete the coloring of the edges of G minimizing the total number of colors used. The problem has been proved to be NP-hard even for bipartite graphs of maximum degree three. Two special cases of the problem have been previously considered and tight upper and ower bounds on the optimal number of colors were proved. The upper bounds led to 3 2 -approximation algorithms for both problems. In this paper we present a randomized (1.37+o(1))-approximation algorithm for the general problem in the case where max{l,c}=ω( ln n) . Our techniques are motivated by recent works on the Circular Arc Coloring problem and are essentially different and simpler than the existing ones.

A better than “best possible” algorithm to edge color multigraphs

By a result of Holyer, unless P = NP, there does not exist a polynomial-time approximation algorithm to edge color a multigraph that always uses fewer than 4 3 χ′ colors, where χ′ is the optimal number of colors. This makes it appear that finding provably good edge colorings is extremely difficult. However, in this paper we present an algorithm to find an edge coloring of a multigraph that never uses more than ⌊ 9 8 χ′ + 3 4 ⌋ colors. In addition, if χ′ ≥ ⌊ 9 8 Δ + 3 4 ⌋ then the algorith optimally colors the graph in polynomial time. Furthermore, this algorithm never uses more than ( 4 3 )χ′ colors and runs in O(| E|(| V| + Δ)) time, where E is the set of edges, and V is the set of vertices.