C/C++ • Re: Boost provides free peer-reviewed portable C++ source libraries.

Pick any vertex v ∈ V as the starting point. ... Since the graph is triangulated, v can have at most 5 neighbors (due to Euler's formula for planar graphs and the handshaking lemma, implying an average degree less than 6, and in a triangulated graph, this can be at most 5 for an internal vertex to maintain planarity).

Guarantee: Given the constraints of planarity and the choice of 6 colors, this algorithm guarantees a valid coloring.

planar verex 6-coloring algorithm

A very specific topic!

Here's an outline of a 6-coloring algorithm for planar graphs, which are graphs that can be embedded in the plane without edge crossings. The famous Four Color Theorem states that every planar graph can be colored with 4 colors, but since you asked for 6, this will be even more straightforward and efficient. I'll provide a step-by-step algorithm, though for simplicity and clarity, it will be more of a conceptual guide rather than a direct implementation in code.

Planar Graph 6-Coloring Algorithm
Input:
A planar graph G = (V, E)
Output:
A 6-coloring of G, i.e., a function color: V → {1, 2, 3, 4, 5, 6} such that for any edge (u, v) ∈ E, color(u) ≠ color(v).
Algorithm:
Preprocessing: Ensure the Graph is Triangulated

If G is not already triangulated (i.e., every face is a triangle), triangulate it by adding edges without violating planarity. This step simplifies the coloring process.
Choose an Arbitrary Vertex: Starting Point

Pick any vertex v ∈ V as the starting point. Color v with color 1.
Color the Neighbors of v Differently

Since the graph is triangulated, v can have at most 5 neighbors (due to Euler's formula for planar graphs and the handshaking lemma, implying an average degree less than 6, and in a triangulated graph, this can be at most 5 for an internal vertex to maintain planarity).
Color each neighbor of v with a different color from the set {2, 3, 4, 5, 6}. If v has fewer than 5 neighbors, some colors will simply not be used for this step.
Iterate Through the Remaining Vertices

Ordering: Establish an ordering of the remaining uncolored vertices based on their distance from v (or simply process them in any order if the graph structure doesn’t immediately suggest a beneficial ordering).
Color Each Vertex:
For each uncolored vertex u:
Determine Available Colors: Identify the colors not used by u's neighbors.
Assign a Color: Since we're working with a triangulated planar graph and have 6 colors at our disposal, and given the previous steps, there will always be at least one available color for u. Assign the smallest available color to u.
Output the Coloring

Once all vertices have been colored, output the coloring function color: V → {1, 2, 3, 4, 5, 6}.
Example Walkthrough:
Consider a simple triangulated planar graph with vertices A, B, C, D, E, F where A is connected to B, C, D; B to C, E; C to D, F; and D, E, F form a triangle.

Start with A, color A with 1.
Color A's Neighbors:
B: 2
C: 3
D: 4
Iterate:
B (already colored 2).
C (already colored 3).
D (already colored 4).
E (neighbors B:2, available colors: {1, 3, 4, 5, 6}, choose 5).
F (neighbors C:3, D:4, available colors: {1, 2, 5, 6}, choose 6).
Result:
A: 1
B: 2
C: 3
D: 4
E: 5
F: 6
Notes:
Efficiency: This algorithm is conceptually simple and efficient for small to medium-sized graphs. For very large graphs, more complex algorithms might offer better performance.
Guarantee: Given the constraints of planarity and the choice of 6 colors, this algorithm guarantees a valid coloring.
Optimization: If minimizing the number of colors is a goal, after coloring with 6, one could attempt to reduce the palette by reassigning colors based on the graph's specific structure, though this would add complexity

Statistics: Posted by HermannSW — Fri Oct 18, 2024 9:44 am