The three-color problem accepts a graph and returns whether it is possible to co

Question

The three-color problem accepts a graph and returns whether it is possible to color each vertex one of 3 different colors so that no adjacent vertices have the same color. For example, the graph to the left below is 3-colorable, and a 3-coloring appears to the right (using colors black, white, and gray Answer the following about developing a backtracking algorithm for the three-color problem. 1. What are the states and transitions in a backtracking algorithm for the three-color problem? 2. How would you design a backtracking algorithm so that it never repeats a previous state? 3. How would you recognize candidates or transitions that cannot lead to a solution?

Explanation / Answer

1. States and transitions in backtracking algorithm for the 3-color problem:

Solution of 3-color problem of a graph with V number of nodes as [v0,...., vV], using backtracking includes the nodes of the graph as states, where, the starting node from the graph will workd as root (the node that will be colored by first color) for state-transition tree and the leaves follows are the states representing the other connected nodes that can be colored by the second color, defining the transition from first to second node, similarly all of the feasible solution(s) (if any) can be represented by space-transition tree.

2. Designing a backtracking algorithm so that it never repeats a previuos state:

Since the backtracking algorithm is for a Graph, to allow the read of a node only once from Graph one can simply use one of the two Depth First Search (DFS) or Breadth First Search (BFS) algorithm, that removes the possibility of repeatedly reading a single state (that is node form the graph).

3. Recongnizing candidates or transitions that cannot lead to a soultion:

It is simple to identify the candidates or transitions that cannot lead a feasible soultion by visualizing the state -transition graphs, where, transitions can also be numbered (or labelled), in accordance to the color plan (say 0, 1, 2...), any candidate are transition leading to the number greater than 2, can be categorized as non-feasible soultion for the 3-coloring problem.

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