Degree-Based Logical Adjacency Checking (DBLAC): A Novel Heuristic for Vertex Coloring

TL;DR

DBLAC introduces a novel vertex coloring heuristic that orders vertices by decreasing degree and uses a new logical AND operation to identify common edges for efficient color propagation. The approach achieves $O(n\Delta)$ time and $O(n+m)$ space, offering competitive or superior color counts and faster runtimes compared to DSATUR and RLF on both randomly generated and DIMACS benchmark graphs. Empirical results demonstrate that DBLAC often reduces the chromatic number by 1–2 colors in benchmark cases and consistently outperforms in runtime, validating its practicality for dense and irregular graphs. The work positions DBLAC as a robust, scalable option for real-world graph coloring tasks with potential applications in scheduling, register allocation, and network design.

Abstract

Degree Based Logical Adjacency Checking (DBLAC). An efficient coloring of graphs with unique logical AND operations. The logical AND operation shows more effective color assignment and fewer number of induced colors in the case of common edges between vertices. In this work, we provide a detailed theoretical analysis of DBLAC's time and space complexity. It furthermore shows its effectiveness through prolonged experiments on standard benchmark graphs. We compare it with existing algorithms, namely DSATUR and Recursive Largest First (RLF). Second, we show how DBLAC achieves competitive results with respect to both the number of colors used and runtime performance.

Figures (2)

• Figure 1: Number of chromatic color
• Figure 2: Time Complexity