Chromatic Number of Bipolar Intuitionistic Fuzzy Graphs

Background Bipolar intuitionistic fuzzy graphs (BIFGs) extend fuzzy graphs by assigning both positive and negative membership and non-membership degrees to vertices and edges, capturing more nuanced uncertainty. The chromatic number of a BIFG, unlike that of crisp graphs, depends on the interplay of these bipolar parameters. This article introduces level graphs and strong level graphs as tools to define and compute the chromatic number of BIFGs. Method We defined a level graph of a BIFG for given thresholds on positive and negative membership values. Using these level graphs, we derive a procedure to assign colors to vertices such that two adjacent vertices have different colors. The chromatic number of the BIFG is then computed as the least number of colors necessary across all valid level graphs. Result The chromatic number of a BIFG is shown to be equal to the maximum chromatic number among its level graphs for all admissible threshold pairs. Several examples illustrate that the proposed level-graph method yields exact chromatic numbers for classes of BIFGs. Conclusion We successfully extended graph coloring to bipolar intuitionistic fuzzy graphs by introducing level graphs and strong level graphs. The chromatic number defined via these structures is well-defined and computable, providing a foundation for further study of BIFG coloring problems. Future work may explore algorithms for efficient determination and applications in bipolar uncertainty-driven scheduling or conflict resolution.