This document provides a detailed solution to the 3-coloring problem using Python code and graph theory concepts.
Extract
[...] Table Result for n = 8 Summit Colour 1 R 2 B 3 R 4 B 5 R 6 B 7 N For , two colors are sufficient: it is enough to make an alternation of two colors in a circular graph. It is important to note here that it is possible to use three colors but the idea is to illustrate that two colors are sufficient and this regardless of the number of vertices , with a positive integer. [...]
[...] The colors chosen, as an example, are: red, blue and black. The test for a number of vertices [...]
[...] The graph problem Question Example of a with six summits and colorables An example of a positive instance of a with six vertices and three colorable can be given as follows: Consider the vertices: and the edges . The following diagram illustrates that the graph has six vertices is well three colorable. Figure Example of a positive instance of a Gp with six vertices three colorable B. Example of a with seven summits and not colorables An example of a negative instance of a with six vertices and three colorable can be given as follows: Consider the vertices: and the edges . The following drawing illustrates that the graph has seven vertices and is not three-colorable. [...]
[...] Each vertex has a color (obvious). 2. Adjacent vertices do not have the same color: necessary but not sufficient condition 3. Four colors are sufficient. [...]
[...] Graphs, Tree Search and Complexity Graphs, Tree Search and Complexity Continuous Control No. 1 1. The Logic Problem Question To answer questions a and the code below, developed in python, has been used: from pysat.solvers import Solver # import the solvers from the pysat library # Define the expressions in the two questions in a two-dimensional list called expressions, the first row corresponds to the first question # The second row corresponds to the second question EXPRESSIONS = [ # First question ( # Clause: ¬a or ¬b or ¬c or ¬d # Second question # Function to check the satisfiability of a formula def check_satisfiability(formula, name): with Solver() as solver: # Ensure the solver is reset at each call for clause in formula: solver.add_clause(clause) # Check if `add_clause` is the correct method satisfiable = solver.solve() print(f'\nResult for the formula if satisfiable: print('The formula is satisfiable.') else: print('The formula is not satisfiable.') # Verify the two formulas check_satisfiability(EXPRESSIONS[0], "Relation check_satisfiability(EXPRESSIONS[1], "Relation In the above code, the variables have been replaced by numbers respectively. [...]
