# PRIMEary Colors

We define the coloring number of a graph of V vertices and E edges to be the smallest number of colors such that no two vertices that share an edge are colored the same. Consider the following graph:

- The vertices are the integers 2, 3, 4, 5, 6, …., 9,998, 9,999, and 10,000
- Two vertices are connected by an edge if they share a common factor. So there isn’t an edge between 2 and 3, but there is between 2 and 6, another between 3 and 6, but none between 2 and 5, between 3 and 5, …

Prove that the coloring number of this graph is at least 13.

Problem created by Steven Miller as a practice test question for Princeton’s Discrete Math 341, Fall ’98.

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