Optical Graph 3-Colorability

The graph 3-colorability problem is a decision problem in graph theory which asks if it is possible to assign a color to each vertex of a given graph using at most three colors, satisfying the condition that every two adjacent vertices have different colors. It has been proved that the graph 3-colorability problem belongs to NP-complete class of problems which no polynomial resources solution is found for them yet. In this paper, a novel optical solution to the graph 3-colorability problem is provided. In this solution, polynomial number of black filters are created in preprocessing phase each of which has exponential size and requires exponential time to be created. After preprocessing phase, the provided solution takes O(n + m) time to decide if a given graph is 3-colorable or not, where the given graph has n vertices and m edges.