TY - JOUR
T1 - Some Results on the Achromatic Number
AU - Cairnie, N.
AU - Edwards, K.
N1 - Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.
PY - 1997
Y1 - 1997
N2 - Let G be a simple graph. The achromatic number ?(G) is the largest number of colors possible in a proper vertex coloring of G 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 () = 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 ?(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 N = N(d, ?) such that if G is a graph with maximum degree at most d, and m = N edges, then (1 - ?)q(m) = ?(G) = q(m).
AB - Let G be a simple graph. The achromatic number ?(G) is the largest number of colors possible in a proper vertex coloring of G 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 () = 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 ?(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 N = N(d, ?) such that if G is a graph with maximum degree at most d, and m = N edges, then (1 - ?)q(m) = ?(G) = q(m).
UR - http://www.scopus.com/inward/record.url?scp=0031517582&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:0031517582
SN - 0364-9024
VL - 26
SP - 129
EP - 136
JO - Journal of Graph Theory
JF - Journal of Graph Theory
IS - 3
ER -