Abstract
The primary aim of this paper is to publicize various problems regarding chromatic coloring of finite, simple and undirected graphs. A simple motivation for this work is that the coloring of graphs gives models for a variety of real world problems such as scheduling. We prove some interesting results related to the computation of chromatic number of certain distance graphs and also discuss some open problems.
Highlights
Grey in [1] found that (G(R2, {1})) is at least 5 by exhibiting a 5-chromatic graph G on 1581 vertices of unit distance. It was further improved by Heule in [2] and his unit distance graph G(R2, {1}) had 553 vertices. What is it that is common between conducting of examination and organizing sports events? It is clear that both the tasks demand huge planning and scheduling
The aim is to determine the least time in which all tasks can be completed
It is known that the task of computation of chromatic number unless it is one or two belongs to the class of NP-complete problems. It means no polynomial time step-by-step procedure could find the chromatic number of any given graph
Summary
What is it that is common between conducting of examination and organizing sports events? It is clear that both the tasks demand huge planning and scheduling. The aim is to determine the least time in which all tasks can be completed We can model this scheduling task with a graph whose vertices denote tasks and the edges among them for conflicts associated with it. The least time required to complete the task corresponds to the problem of computing the chromatic number of the respective graph. It is known that the task of computation of chromatic number unless it is one or two belongs to the class of NP-complete problems. It means no polynomial time step-by-step procedure could find the chromatic number of any given graph. By the famous 4-color Theorem of graph colorings, we require only 4 distinct channels to assure this constraint independent of the cell shape
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