Add Graph Colouring algorithm with Backtracking

Author: AswinPKumar01

graphs/graph_colouring.py

@@ -0,0 +1,83 @@
+"""
+Graph Coloring Problem is a classic problem in graph theory.
+The task is to assign colors to the vertices of a graph so that no two adjacent vertices
+share the same color, and the number of colors used is minimized.
+
+Wikipedia: https://en.wikipedia.org/wiki/Graph_coloring
+"""
+
+
+def is_safe(graph: list[list[int]], color: list[int], v: int, c: int) -> bool:
+    """
+    Helper function to check if it is safe to color vertex `v` with color `c`.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    color (list[int]): The list of colors assigned to each vertex.
+    v (int): The vertex to check.
+    c (int): The color to be assigned.
+
+    Returns:
+    bool: True if it's safe to assign color `c` to vertex `v`, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> color = [-1, -1, -1]
+    >>> is_safe(graph, color, 0, 1)
+    True
+    """
+    return all(not (graph[v][i] == 1 and color[i] == c) for i in range(len(graph)))
+
+
+def graph_coloring_util(
+    graph: list[list[int]], m: int, color: list[int], v: int
+) -> bool:
+    """
+    Utility function that uses backtracking to solve the m-coloring problem.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    m (int): The maximum number of colors.
+    color (list[int]): The list of colors assigned to each vertex.
+    v (int): The current vertex to be colored.
+
+    Returns:
+    bool: True if all vertices are successfully colored, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> color = [-1, -1, -1]
+    >>> graph_coloring_util(graph, 3, color, 0)
+    True
+    """
+    if v == len(graph):
+        return True
+
+    for c in range(1, m + 1):
+        if is_safe(graph, color, v, c):
+            color[v] = c
+            if graph_coloring_util(graph, m, color, v + 1):
+                return True
+            color[v] = -1  # Backtrack
+
+    return False
+
+
+def graph_coloring(graph: list[list[int]], m: int) -> bool:
+    """
+    Solves the m-coloring problem using backtracking.
+
+    Parameters:
+    graph (list[list[int]]): The adjacency matrix of the graph.
+    m (int): The maximum number of colors.
+
+    Returns:
+    bool: True if the graph can be colored with `m` colors, otherwise False.
+
+    Example:
+    >>> graph = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
+    >>> graph_coloring(graph, 3)
+    True
+    """
+    color = [-1] * len(graph)
+    return graph_coloring_util(graph, m, color, 0)