Improve comments, add doctests to coloring.py (#11589)

* Improve comments, add doctests to coloring.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Fix ruff errors

* Update backtracking/coloring.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update backtracking/coloring.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update backtracking/coloring.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update backtracking/coloring.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update backtracking/coloring.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Remove type hints for function docstrings

* updating DIRECTORY.md

* updating DIRECTORY.md

* pyproject.toml: tool.rumdl.per-file-ignores."index.md" = ["MD040"]

* Clean up toctree in index.md

Removed unnecessary comment from the toctree directive.

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: Hardvan <Hardvan@users.noreply.github.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: 3 people

DIRECTORY.md

@@ -752,6 +752,7 @@
   * [Decimal Isolate](maths/decimal_isolate.py)
   * [Decimal To Fraction](maths/decimal_to_fraction.py)
   * [Derangement](maths/derangement.py)
+  * [Digital Root](maths/digital_root.py)
   * [Dodecahedron](maths/dodecahedron.py)
   * [Double Factorial](maths/double_factorial.py)
   * [Dual Number Automatic Differentiation](maths/dual_number_automatic_differentiation.py)

backtracking/coloring.py

@@ -1,7 +1,7 @@
 """
-Graph Coloring also called "m coloring problem"
-consists of coloring a given graph with at most m colors
-such that no adjacent vertices are assigned the same color
+Graph Coloring (also called the "m coloring problem") is the problem of
+assigning at most 'm' colors to the vertices of a graph such that
+no two adjacent vertices share the same color.
 
 Wikipedia: https://en.wikipedia.org/wiki/Graph_coloring
 """
@@ -11,22 +11,46 @@ def valid_coloring(
     neighbours: list[int], colored_vertices: list[int], color: int
 ) -> bool:
     """
+    Check if a given vertex can be assigned the specified color
+    without violating the graph coloring constraints (i.e., no two adjacent vertices
+    have the same color).
+
+    Procedure:
     For each neighbour check if the coloring constraint is satisfied
     If any of the neighbours fail the constraint return False
     If all neighbours validate the constraint return True
 
-    >>> neighbours = [0,1,0,1,0]
+    Parameters:
+    neighbours: The list representing which vertices
+                are adjacent to the current vertex.
+                1 indicates an edge between the current vertex
+                and the neighbour.
+    colored_vertices: List of current color assignments for all vertices
+                      (-1 means uncolored).
+    color: The color we are trying to assign to the current vertex.
+
+    Returns:
+    True if the vertex can be safely colored with the given color,
+    otherwise False.
+
+    Examples:
+    >>> neighbours = [0, 1, 0, 1, 0]
     >>> colored_vertices = [0, 2, 1, 2, 0]
-
     >>> color = 1
     >>> valid_coloring(neighbours, colored_vertices, color)
     True
 
     >>> color = 2
     >>> valid_coloring(neighbours, colored_vertices, color)
     False
+
+    >>> neighbors = [1, 0, 1, 0]
+    >>> colored_vertices = [-1, -1, -1, -1]
+    >>> color = 0
+    >>> valid_coloring(neighbors, colored_vertices, color)
+    True
     """
-    # Does any neighbour not satisfy the constraints
+    # Check if any adjacent vertex has already been colored with the same color
     return not any(
         neighbour == 1 and colored_vertices[i] == color
         for i, neighbour in enumerate(neighbours)
@@ -37,7 +61,7 @@ def util_color(
     graph: list[list[int]], max_colors: int, colored_vertices: list[int], index: int
 ) -> bool:
     """
-    Pseudo-Code
+    Recursive function to try and color the graph using backtracking.
 
     Base Case:
     1. Check if coloring is complete
@@ -51,6 +75,20 @@ def util_color(
             2.4. if current coloring leads to a solution return
             2.5. Uncolor given vertex
 
+    Parameters:
+    graph: Adjacency matrix representing the graph.
+           graph[i][j] is 1 if there is an edge
+           between vertex i and j.
+    max_colors: Maximum number of colors allowed (m in the m-coloring problem).
+    colored_vertices: Current color assignments for each vertex.
+                      -1 indicates that the vertex has not been colored
+                      yet.
+    index: The current vertex index being processed.
+
+    Returns:
+    True if the graph can be colored using at most max_colors, otherwise False.
+
+    Examples:
     >>> graph = [[0, 1, 0, 0, 0],
     ...          [1, 0, 1, 0, 1],
     ...          [0, 1, 0, 1, 0],
@@ -67,43 +105,68 @@ def util_color(
     >>> util_color(graph, max_colors, colored_vertices, index)
     False
     """
-
-    # Base Case
+    # Base Case: If all vertices have been assigned a color, we have a valid solution
     if index == len(graph):
         return True
 
-    # Recursive Step
-    for i in range(max_colors):
-        if valid_coloring(graph[index], colored_vertices, i):
-            # Color current vertex
-            colored_vertices[index] = i
-            # Validate coloring
+    # Try each color for the current vertex
+    for color in range(max_colors):
+        # Check if it's valid to color the current vertex with 'color'
+        if valid_coloring(graph[index], colored_vertices, color):
+            colored_vertices[index] = color  # Assign color
+            # Recur to color the rest of the vertices
             if util_color(graph, max_colors, colored_vertices, index + 1):
                 return True
-            # Backtrack
+            # Backtrack if no solution found with the current assignment
             colored_vertices[index] = -1
-    return False
+
+    return False  # Return False if no valid coloring is possible
 
 
 def color(graph: list[list[int]], max_colors: int) -> list[int]:
     """
-    Wrapper function to call subroutine called util_color
-    which will either return True or False.
-    If True is returned colored_vertices list is filled with correct colorings
-
+    Attempt to color the graph with at most max_colors colors such that no two adjacent
+    vertices have the same color.
+    If it is possible, returns the list of color assignments;
+    otherwise, returns an empty list.
+
+    Parameters:
+    graph: Adjacency matrix representing the graph.
+    max_colors: Maximum number of colors allowed.
+
+    Returns:
+    List of color assignments if the graph can be colored using max_colors.
+    Each index in the list represents the color assigned
+    to the corresponding vertex.
+    If coloring is not possible, returns an empty list.
+
+    Examples:
     >>> graph = [[0, 1, 0, 0, 0],
     ...          [1, 0, 1, 0, 1],
     ...          [0, 1, 0, 1, 0],
     ...          [0, 1, 1, 0, 0],
     ...          [0, 1, 0, 0, 0]]
-
     >>> max_colors = 3
     >>> color(graph, max_colors)
     [0, 1, 0, 2, 0]
 
     >>> max_colors = 2
     >>> color(graph, max_colors)
     []
+
+    >>> graph = [[0, 1], [1, 0]]  # Simple 2-node graph
+    >>> max_colors = 2
+    >>> color(graph, max_colors)
+    [0, 1]
+
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]  # Complete graph of 3 vertices
+    >>> max_colors = 2
+    >>> color(graph, max_colors)
+    []
+
+    >>> max_colors = 3
+    >>> color(graph, max_colors)
+    [0, 1, 2]
     >>> color([], 2)  # empty graph
     []
     >>> color([[0]], 1)  # single node, 1 color
@@ -113,9 +176,11 @@ def color(graph: list[list[int]], max_colors: int) -> list[int]:
     >>> color([[0, 1], [1, 0]], 2)  # 2 nodes, 2 colors (possible)
     [0, 1]
     """
+    # Initialize all vertices as uncolored (-1)
     colored_vertices = [-1] * len(graph)
 
+    # Use the utility function to try and color the graph starting from vertex 0
     if util_color(graph, max_colors, colored_vertices, 0):
-        return colored_vertices
+        return colored_vertices  # The successful color assignment
 
-    return []
+    return []  # No valid coloring is possible

index.md

@@ -1,10 +1,10 @@
 # TheAlgorithms/Python
 
-```text {toctree}
+<!-- CONTRIBUTING.md must be the FIRST doc and README.md can come after. -->
+```{toctree}
 :maxdepth: 2
 :caption: index.md
 
-<!-- CONTRIBUTING.md must be the FIRST doc and README.md can come after. -->
 CONTRIBUTING.md
 README.md
 LICENSE.md

pyproject.toml

@@ -242,3 +242,4 @@ disable = [
 exclude = [ "DIRECTORY.md", "docs/hacktober_2026_prep.md" ]
 flavor = "gfm"
 line-length = 350
+per-file-ignores."index.md" = [ "MD040" ]