Some results on the achromatic number

Let G be a simple graph. The achromatic number ψ(G) is the largest number of colors possible in a proper vertex coloring ofG in which each pair of colors is adjacent somewhere in G. For any positive integer m, let q(m) be the largest integer k such that (2 ) ≤ m. We show that the problem of determining the achromatic number of a tree is NP-hard. We further prove that almost all trees T satisfy ψ(T ) = q(m), where m is the number of edges in T . Lastly, for fixed d and > 0, we show that there is an integer N0 = N0(d, ) such that if G is a graph with maximum degree at most d, and m ≥ N0 edges, then (1 − )q(m) ≤ ψ(G) ≤ q(m). c © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 129–136, 1997