-- Application of Groebner basis to determining whether a given planar non-cyclical
-- graph can be colored with 3 colors where neighboring vertices (connected by a single edge)
-- have different colors

-- We express the problem in terms of polynomials. Assign the three distinct cuve roots of 
-- unity to represent each color then for n vertices we have n equations in the ideal I of the form:
--       X_i ^ 3 - 1 = 0     (X_i represents the vertices since each vertex must be colored
--       these polynomials are in the ideal)
-- The vertices X_i, X_j connected by an edge must have a different color
-- since X_i^3 = X_j^3 [1] we must have that (X_i - X_j)(X_i^2 + X_i*X_j + X_j^2)=0 [2]
-- because the two polynomials are equivalent it must be that (X_i^2 + X_i*X_j + X_j^2)=0
-- for all X_i, X_j that are neighbors. These too are in the ideal I. 

-- Theorem: A graph F is 3 colorable if and only if V(I) is nonempty.
-- In other words if there is a common solution to all the polynomials that define a graph
-- then it is 3 colorable.

-- We us Groebner basis to determine whether or not the variety formed by the above ideal
-- is empty. It is if 1 belongs to the resulting Groebner basis


isTriColorable = (vertices, vertexList)-> (
     	       	    R=QQ[z_1..z_(#vertices+1), MonomialOrder=>Lex];
		    i=1;
		    j=0;
		    K={};	      
		    while i < #vertices+1 do (
			 K=append(K,(z_i)^3-1);			
			 i=i+1;
		   );
	      	   while j < #vertexList do (
			K=append(K,z_(vertexList#j#0)^2 + z_(vertexList#j#0)*z_(vertexList#j#1) + z_(vertexList#j#1)^2);					
			j=j+1;
		       );
		   I=ideal(K); 
		   G= gb(I);	       
	      	   return G;
		   
		   )
	      
--Examples
--Simple graph, with only one way to color (up to permutation of color) so the Groebner Basis
--is also simple	      
example1vertices={1,2,3,4,5,6,7,8};
example1graph={{1,2},{1,5},{1,6},{2,3},{2,4},{2,8},{3,4},{3,8},{4,5},{4,7},{5,6},{5,7},{6,7},{7,8}};
Ex1= time isTriColorable(example1vertices, example1graph)
--Same graph as above except with one extra edge
notcolorable = {1,2,3,4,5,6,7,8};
notcolorablegraph ={{1,2},{1,5},{1,6},{2,3},{2,4},{2,5},{2,8},{3,4},{3,8},{4,5},{4,7},{5,6},{5,7},{6,7},{7,8}};
Ex2 = isTriColorable(notcolorable, notcolorablegraph)
--Final test
1 % Ex2 == 0
1 % Ex1 == 0
M1 = transpose gens Ex1
degree ideal gens Ex1
--Graph that can be colored several different ways, complicated Groebner basis takes longer to compute
example3vertices={1,2,3,4,5,6,7,8,9};
example3graph = {{1,2},{1,4},{1,5},{1,6},{2,3},{2,5},{2,7},{3,4},{3,6},{3,9},{5,6},{6,8},{7,8},{8,9}};
Ex3 = time isTriColorable(example3vertices, example3graph)
M3 = transpose gens Ex3
dim ideal gens Ex3
degree ideal gens Ex3
degree radical ideal gens Ex3
1 % Ex3 == 0

-- This allows to tell whether or not any planar graph is 3 colorable. Theorems about
-- 3 colorability of graphs refer to specific graphs such as triangluated graphs in which
-- all vertices have an even degree are always 3 colorable. Or the Grotzch's theorem which
-- states that any triangle free (no 3 cycle) graph is 3 colorable

-- The End
-- Special bonus Four colorable, just the same except that in place 
-- of cube roots of unity we have quadratic roots, and 
-- the polynomial of neighboring vertices, should be
--       x_i^3 + x_i^2*x_j + x_i*x_j + x_j^3 = 0



isFourColorable = (vertices, vertexList)-> (
     	       	    R=QQ[z_1..z_(#vertices+1), MonomialOrder=>Lex];
		    i=1;
		    j=0;
		    K={};	      
		    while i < #vertices+1 do (
			 K=append(K,(z_i)^4-1);			
			 i=i+1;
		   );
	      	   while j < #vertexList do (
			K=append(K,z_(vertexList#j#0)^3 + z_(vertexList#j#0)^2*z_(vertexList#j#1)
             + z_(vertexList#j#0)*z_(vertexList#j#1)^2+ z_(vertexList#j#1)^3);					
			j=j+1;
		       );
		   I=ideal(K); 
		   G= gb(I);	       
	      	   return G;
		   
		   )

fourcolorable = {1,2,3,4,5,6,7,8};
fourcolorablegraph ={{1,2},{1,5},{1,6},{2,3},{2,4},{2,5},{2,8},{3,4},{3,8},{4,5},{4,7},{5,6},{5,7},{6,7},{7,8}};
gbTrace 3
Ex2 =trac isFourColorable(fourcolorable, fourcolorablegraph)
--Final test
1 % Ex2 == 0