Randomized Algorithm for Finding a Coloring with at Most Cn^4/64 Monochromatic K4s

This article presents a randomized algorithm for finding a coloring of a graph with at most Cn^4/64 monochromatic copies of K4, where C is a constant, that runs in expected time polynomial in n.

The algorithm proceeds as follows:

1. Random Graph Generation: Generate a random graph G with n vertices.

2. Random Coloring: Assign each vertex in G a random color from {1, 2, 3, 4}.

3. Monochromatic K4 Detection and Removal: For each set of 4 vertices in G, check if they form a monochromatic K4. If they do, remove this monochromatic K4 from G.

4. Iteration: Repeat steps 2 and 3 until no more monochromatic K4s exist in G.

5. Output: If G is a valid colored graph, return G. Otherwise, return to step 1.

Since the algorithm is randomized, its expected running time is polynomial in n. Moreover, as each check results in the removal of a monochromatic K4, the algorithm performs at most Cn^4/64 checks, guaranteeing a maximum of Cn^4/64 monochromatic K4s in the final coloring. This ensures the desired bound on the number of monochromatic K4s.