Quantum Algorithms for Colouring of Graphs

Colouring of graphs is a fundamental aspect of graph theory with numerous applications in computer science, including scheduling, map coloring, and network design. Despite existing methods for determining graph colorability, there is a notable absence of quantum algorithms specifically aimed at both assessing a graph's colorability and performing the coloring task itself. This study presents a novel quantum algorithm leveraging the principle of quantum entanglement to perform 2- and 3-vertex coloring of graphs. The algorithm synthesizes basic quantum gates into quantum circuits that first ascertain whether a given graph is colorable with 2 or 3 colors, followed by executing the necessary operations to effectuate the coloring process. Our findings demonstrate the feasibility of employing quantum resources to tackle graph coloring problems, which could significantly enhance efficiency in computer science-related tasks that require graph colorability analysis. The results emphasize the potential of quantum computing to transform approaches in solving complex problems traditionally addressed by classical algorithms.