Introduction
Imagine trying to schedule multiple classes at a school so that no two classes sharing the same teacher happen at the same time. Or assigning frequencies to cell towers so that no neighboring towers interfere. This is where Graph Colouring comes in—a classic but notoriously difficult problem in computer science.
What is Graph Colouring?
At its core, graph colouring involves assigning colors to the vertices of a graph such that no two adjacent vertices share the same color.
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A vertex is like a point or node.
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An edge connects two vertices.
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A valid coloring means adjacent vertices get different colors.
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The chromatic number is the minimum number of colors needed to achieve this.
Classic Backtracking: The Smart Brute Force
The traditional way to tackle graph colouring is through backtracking. It’s a recursive method where you assign colors to one vertex at a time. If you hit a conflict (i.e., no valid color is left for a vertex), you go back to the previous step and try a different color.
Backtracking Algorithm
def is_safe(graph, v, color, c):
for i in range(len(graph)):
if graph[v][i] == 1 and color[i] == c:
return False
return True
def graph_coloring(graph, m, color, v):
if v == len(graph):
return True
for c in range(1, m+1):
if is_safe(graph, v, color, c):
color[v] = c
if graph_coloring(graph, m, color, v+1):
return True
color[v] = 0 # Backtrack
return False
Backtracking is powerful because it avoids many dead ends early—but as the graph gets complex.
The Quantum-Inspired Twist
Instead of checking one color at a time, quantum-inspired coloring acts like it's checking many color options at once—just like how quantum systems work.
It borrows ideas like:
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Superposition – trying many color choices in parallel
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Smart guessing – choosing colors based on what’s likely to work
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Flexible switching – quickly changing colors if something doesn’t fit
It’s not actual quantum computing, but it works as if it's exploring multiple solutions together, helping solve the problem faster.
1. Quantum-Inspired 2-Coloring of Graphs
What is 2-coloring?
Assigning just
two colors to graph vertices so that
no two connected vertices share the same color.
Which graphs are 2-colorable?
Trees
Even cycles
2D grids
Linear graphs with even nodes
How does the quantum circuit work?
1) All qubits start in the |0⟩ state.
2) Use anti-controlled NOT gates to flip every alternate qubit to simulate alternating colors.(like red and blue).
3) Final states alternate like 010101... or 101010....
2-coloring of linear graph
Quantumcircuit for 2-coloring
2. Quantum-Inspired 3-Coloring of Graphs
What is 3-coloring? Coloring a graph with three distinct colors so that no two connected vertices have the same color.
Quantum strategy:
1) Uses entangled states to represent connections (edges).
2) Extra helper qubits (ancilla) store information about these connections.
3) Controlled gates are applied based on connection data to assign color operations.
Color Representation:
Identity gate (I) = Red
NOT gate (X) = Green
Hadamard gate (H) = Blue
How does it decide coloring?
a) Checks which vertices are connected using ancilla qubits.
b) Based on different connection combinations, controlled operations apply different color gates.
3-coloring of linear graph
Quantumcircuit for 2-coloring
Conclusion
Quantum-inspired graph coloring offers a creative way to tackle classic problems. By simulating principles like superposition and entanglement, it explores many coloring options simultaneously—much faster than traditional methods.
It efficiently handles 2-coloring and 3-coloring, with applications in scheduling, map coloring, timetabling, and more. As quantum computing advances, these ideas could soon power smarter real-world tools.
References
1. "Quantum Algorithms for Graph Coloring" - Go here to read it : [
Research Paper].
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