update docs

Author: GiggleLiu

examples/Matching.jl

@@ -69,4 +69,6 @@ matching_poly = solve(problem, GraphPolynomial())[]
 match_config = solve(problem, SingleConfigMax())[]
 
 # Let us show the result by coloring the matched edges to red
-show_graph(graph; locs=locations, edge_colors=[isone(match_config.c.data[i]) ? "red" : "black" for i=1:ne(graph)])
+show_graph(graph; locs=locations, edge_colors=[isone(match_config.c.data[i]) ? "red" : "black" for i=1:ne(graph)])
+
+# where we use edges with red color to related pairs of matched vertices.

examples/MaxCut.jl

@@ -72,4 +72,6 @@ max_cut_size_verify = cut_size(graph, max_vertex_config)
 # You should see a consistent result as above `max_cut_size`.
 
 show_graph(graph; locs=locations, vertex_colors=[
-        iszero(max_vertex_config[i]) ? "white" : "red" for i=1:nv(graph)])
+        iszero(max_vertex_config[i]) ? "white" : "red" for i=1:nv(graph)])
+
+# where red vertices and white vertices are seperated by the cut.

examples/PaintShop.jl

@@ -32,7 +32,8 @@ for i=1:length(sequence)
     i != j && add_edge!(graph, i, j)
 end
 
-show_graph(graph; locs=locations, texts=string.(sequence), edge_colors=[sequence[e.src] == sequence[e.dst] ? "blue" : "black" for e in edges(graph)])
+show_graph(graph; locs=locations, texts=string.(sequence), edge_colors=
+    [sequence[e.src] == sequence[e.dst] ? "blue" : "black" for e in edges(graph)])
 
 # Vertices connected by blue edges must have different colors,
 # and the goal becomes a min-cut problem defined on black edges.
@@ -80,25 +81,22 @@ max_config = solve(problem, GraphPolynomial())[]
 # ```math
 # D(G, x) = \sum_{k=0}^{\delta(G)} d_k x^k 
 # ```
-# where ``2d_k`` is the number of possible coloring with number of color changes ``2m-1-k``.
+# where ``d_k`` is the number of possible coloring with number of color changes ``2m-1-k``.
 paint_polynomial = solve(problem, GraphPolynomial())[]
 
 # ### Configuration properties
 # ##### finding best solutions
 best_configs = solve(problem, ConfigsMax())[]
 
-painting1 = paint_shop_coloring_from_config(best_configs.c.data[1]; initial=false)
-
-show_graph(graph; locs=locations, texts=string.(sequence), edge_colors=[sequence[e.src] == sequence[e.dst] ? "blue" : "black" for e in edges(graph)],
-                vertex_colors=[isone(c) ? "red" : "black" for c in painting1], vertex_text_color="white")
+# One can see to identical bitstrings corresponding two different vertex configurations, they are related to bit-flip symmetry.
 
-#
+painting1 = paint_shop_coloring_from_config(best_configs.c.data[1]; initial=false)
 
-painting2 = paint_shop_coloring_from_config(best_configs.c.data[2]; initial=false)
+show_graph(graph; locs=locations, texts=string.(sequence),
+    edge_colors=[sequence[e.src] == sequence[e.dst] ? "blue" : "black" for e in edges(graph)],
+    vertex_colors=[isone(c) ? "red" : "black" for c in painting1], vertex_text_color="white")
 
-show_graph(graph; locs=locations, texts=string.(sequence), edge_colors=[sequence[e.src] == sequence[e.dst] ? "blue" : "black" for e in edges(graph)],
-                vertex_colors=[isone(c) ? "red" : "black" for c in painting2], vertex_text_color="white")
+# Since we have different choices of initial color, the number of best solution is 2.
 
-# Since we have different choices of initial color, the number of best solution is 4.
 # The following function will check the solution and return you the number of color switchs
 num_paint_shop_color_switch(sequence, painting1)