A coloring of a graph G is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees T₁ and T₂ equals min{Δ(T₁) + 1, Δ(T₂) + 1}. We also prove that the acyclic chromatic number of direct product of two complete graphs Kₘ and Kₙ is mn-m-2, where m ≥ n ≥ 4. Several bounds for the acyclic chromatic number of direct products are given and in connection to this some questions are raised.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1408,
author = {Simon \v Spacapan and Aleksandra Tepeh Horvat},
title = {On acyclic colorings of direct products},
journal = {Discussiones Mathematicae Graph Theory},
volume = {28},
year = {2008},
pages = {323-333},
zbl = {1156.05018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1408}
}
Simon Špacapan; Aleksandra Tepeh Horvat. On acyclic colorings of direct products. Discussiones Mathematicae Graph Theory, Tome 28 (2008) pp. 323-333. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1408/
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