Concerning the achromatic number of graphs
Journal of Combinatorial Theory Series B
Achromatic number is NP-complete for cographs and interval graphs
Information Processing Letters
Introduction to algorithms
Handbook of combinatorics (vol. 1)
Handbook of combinatorics (vol. 1)
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Some results on the achromatic number
Journal of Graph Theory
The achromatic number of bounded degree trees
Discrete Mathematics
Approximation algorithms for the achromatic number
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
On approximating the achromatic number
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Efficient Approximation Algorithms for the Achromatic Number
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Lecture notes on approximation algorithms: Volume I
Lecture notes on approximation algorithms: Volume I
Extremal Graph Theory
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The achromatic number problem is, given a graph G = (V, E), to find the greatest number of colors, Ψ(G), in a coloring of the vertices of G such that adjacent vertices get distinct colors and for every pair of colors some vertex of the first color and some vertex of the second color are adjacent. This problem is NP-complete even for trees. We obtain the following new results using combinatorial approaches to the problem. (1) A polynomial time O(|V|3/8)-approximation algorithm for the problem on graphs with girth at least six. (2) A polynomial time 2-approximation algorithm for the problem on trees. This is an improvement over the best previous 7-approximation algorithm. (3) A linear time asymptotic 1.414-approximation algorithm for the problem when graph G is a tree with maximum degree d(|V|), where d : N → N, such that d(|V|) = O(Ψ(G)). For example, d(|V|) = Θ(1) or d(|V|) = Θ(log |V|). (4) A linear time asymptotic 1.118-approximation algorithm for binary trees. We also improve the lower bound on the achromatic number of binary trees.