On chromatic number and perfectness of fuzzy graph

Abstract

Perfect graphs have a significant role in graph theory in terms of structure as well as application. This article embarked a comprehensive study on perfect fuzzy graphs attempted to obtain the fuzzy analog of the famous weak and strong perfect graph theorems following due classification of fuzzy graphs in terms of perfectness. The inter-relationship between chromatic numbers of a fuzzy graph and its complement graph has been studied in detail deriving important theories and results. It has been shown that unlike crisp case, for fuzzy graphs there exists one and only one perfect fuzzy graph theorem satisfying both the conditions of strong and weak perfect graph theorems. A detailed study of the perfectness of different types of fuzzy graphs has been carried out comparing the results with those obtained on crisp graphs. It has been shown through illustrative examples how the perfectness of a fuzzy graph and that of its underlying crisp graph is independent of each other. Finally, the use of perfect fuzzy graph on application grounds has been evaluated by using it in analyzing how a huge percentage of population in a city is unknowingly living under a big threat due exposure to electromagnetic radiation emitted from cell-phone towers.