Zykov sums of digraphs with diachromatic number equal to its harmonious number

Abstract

The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising from the Zykov sum of digraphs that accept a complete k -coloring with k = 1+ √ 1+4 m 2 for a suitable m . As a consequence, the diachromatic number equals the harmonious number for every digraph in this family. In particular, we determine the diachromatic number of digraphs arising from the Zykov sum of Hamiltonian factorizations of complete digraphs over a suitable digraph. We also obtain the equivalent results for graphs. Furthermore, we determine the achromatic number for digraphs arising from the generalized composition in terms of the thickness of complete graphs. Finally, we extend some results on the dichromatic number of Zykov sums of tournaments to the class of digraphs that are not tournaments and apply them, and the results obtained for the diachromatic number, to the problem of the existence of a digraph with dichromatic number r and diachromatic number t for some particular cases with 2 ≤ r ≤ t .