In the Euclidean k-means problems we are given as input a set of n points in ℝ^d and the goal is to find a set of k points C ⊆ ℝ^d, so as to minimize the sum of the squared Euclidean distances from each point in P to its closest center in C. In this paper, we formally explore connections between the k-coloring problem on graphs and the Euclidean k-means problem. Our results are as follows:

- For all k ≥ 3, we provide a simple reduction from the k-coloring problem on regular graphs to the Euclidean k-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean 2-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem.

- In the other direction, we mimic the O(1.7297ⁿ) time algorithm of Williams [TCS'05] for the max-cut of problem on n vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in 2ⁿ⋅ poly(n,d) time.

- We prove similar results and connections as above for the Euclidean k-min-sum problem.