Let G be a legally colored graph with k colors; this means that each two adjacent vertices have different colors, and the total number of colors in G is k.
In addition, the edges of the graph are colored so that the color of the edge is similar to one of the color of the two vertices it connects.
I am supposed to concoct a formula in propositional logic for coloring the edges of the graph, given the formula for coloring the vertices, however I am confused in doing the proper formulas for the edges. So the questions are:
(1) Which atomic statements should be added to those I already defined?
(2) Which statements should I add to depict the edge coloring of the graph? I thought of this:
T= {$(P_u^i,P_w^j )^S | s⋲ [i,j], u,w⋲V,(u,w)⋲E $}
Vertices coloring -
The number in the bottom is the number of the vertices, and the "i" depicts the color of the vertices
This means that the vertices at least have one color
This means that each vertices has only one color
This means that connected vertices have different colors