Given a graph = ( , ), a coloring function assigns an integer value ( ) to each node in such a way that the extremes of any edge { , } cannot share the same color, i.e., ( ) ≠ ( ). The classical concept of the (crisp) chromatic number of a graph is generalized to fuzzy concept in this paper. Main approach is based on the successive coloring functions of the crisp graphs = ( ; ), the −cuts of ; the traffic lights problem is analyzed following this approach.

Abstract. Concepts of graph theory are applied in many areas of computer science including image segmentation, data mining, clustering, image capturing and networking. Fuzzy graph theory is successfully used in many problems, to handle the uncertainty that occurs in graph theory. An interval-valued fuzzy graph is a generalized structure of a fuzzy graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, new concepts of irregular interval-valued fuzzy graphs such as neighbourly totally irregular intervalvalued fuzzy graph, highly irregular interval-valued fuzzy graphs and highly totally irregular interval-valued fuzzy graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular intervalvalued fuzzy graphs are equivalent is discussed.

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