Chromatic Number of Fuzzy Graphs: Operations, Fuzzy Graph Coloring, and Applications

We focus on fuzzy graphs with crisp vertex sets and fuzzy edge sets. This paper introduces a new concept of chromatic number (crisp) for a fuzzy graph G˜(V,E˜). Moreover, we define the operations of cap, join, difference, ring sum, direct product, semiproduct, strong product, and Cartesian product of fuzzy graphs. Furthermore, the exact value or the upper boundary of the chromatic number of these fuzzy graphs is obtained based on the α-cuts of G˜. Finally, two applications of the chromatic number to solve the timetabling problem and the traffic light problem are analyzed.