Dominator Sum Coloring in Triangular Snake Graphs Optimization Approach using Algorithms and Python Implementation

Dominator coloring is an important area of graph theory with applications and coloring methods often overlook the dominance relationships between vertex color classes, leading to inefficiencies. This paper introduces dominator sum coloring, a novel hybrid approach combining dominator coloring and sum coloring to address these inefficiencies. The goal of this study is to assign labels to vertices that minimize the sum of labels while ensuring each vertex dominates at least one-color class, thereby reducing the number of color classes needed. We apply dominator sum coloring to triangular snake graph families, including triangular snake Tn, double triangular snake DTn, triple triangular snake TTn. Through algorithmic techniques, we compute the dominator sum chromatic number, χds(G) and the chromatic sum Sds(G) graph. for these graphs. Our results show that the dominator sum chromatic number significantly reduces the number of color classes required compared to traditional dominator coloring. Additionally, the chromatic sum is minimized, leading to a more efficient allocation of vertex labels. This paper compares dominator sum coloring within dominator coloring methods, focusing on computational efficiency and real-world applications. The proposed approach outperforms conventional methods in terms of both computational complexity and optimization. It offers an efficient solution for managing dominance relationships in graphs, with practical applications in wireless network design, social network analysis, and resource allocation, where optimizing dominance can improve overall performance.