Optimal Total Coloring Techniques for Enhancing Honeycomb Network Performance

Honeycomb networks—also referred to as honey graphs, owing to their hexagonal lattice representation in graph theory are among the most influential topological structures used in computational and engineering systems. Their geometric regularity, scalability, and symmetry make them fundamental in domains such as parallel processing, VLSI layout design, chemical molecular modeling, wireless communication, and distributed computing. In these systems, efficient resource allocation and conflict-free scheduling are critical for performance optimization. Total coloring, which assigns colors to both vertices and edges such that no adjacent or incident elements share a color, provides a robust mathematical tool for enforcing these operational constraints. Equitable total coloring further requires that color class sizes differ by at most one, a property essential for fairness in load-balanced systems. Although extensive work exists on total coloring, equitable total coloring of honey graphs remains mostly unexplored, particularly in relation to optimal bounds and constructive algorithms. This study determines the equitable total chromatic number of honey graphs of varying orders and develops constructive algorithms that ensure balanced color classes. Results show that the equitable total chromatic number conforms to Δ + 1 or Δ + 2, consistent with the Total Coloring Conjecture, and equitability does not increase the chromatic requirement. These findings confirm that honey graphs inherently support balanced total colorings suitable for conflict-free and fairness-driven scheduling in complex networks.