Maximizing the Chances of a Color Match—Web Supplement

Proof. Let Km denote the complete graph on m vertices. Given an n-color list for each vertex of Km, let p denote the probability of obtaining a match. We will prove that having identical lists minimizes p̄ = 1− p. To compute p̄, we divide the number of proper colorings by the total number of ways to color the vertices. The latter is just n, regardless of whether or not the lists are identical. So it’s enough to show that the number of proper colorings is minimized when all the lists are identical. If n < m, i.e., there are more vertices than colors, then a match cannot be avoided when all the lists are identical, since all the vertices are adjacent to each other. So the number of proper colorings is zero, and we are done. So we’ll assume n ≥ m. Let’s try to count the number of proper colorings for an arbitrary set of n-color lists. Denote the vertices of Km by v1, · · · , vm. There are n ways to choose a color for v1. Having picked colors for v1, · · · , vk, where k ∈ {1, · · · , n− 1}, there are at least n− k ways to pick a color for vk+1. If all the lists are identical, then this lower bound of n− k is in fact achieved, because: (1) all the vertices are adjacent; so (2) v1, · · · , vk must have k distinct colors; and (3) these k colors appear in the list for vk+1. Therefore there are always at least n(n− 1) · · · (n−m + 1) proper colorings for Km, and this lower bound is achieved if all the lists are identical. THEOREM 3. Every tree is n-monophilic for all n ≥ 2.