Abstract
Let G be a graph of maximum degree Δ. A proper vertex coloring of G is acyclic if there is no bichromatic cycle. It was proved by Alon et al. [Acyclic coloring of graphs. Random Structures Algorithms, 1991, 2(3): 277–288] that G admits an acyclic coloring with O(Δ4/3) colors and a proper coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic for a fixed integer k ≥ 5. In this paper, we combine above two colorings and show that if k ≥ 5 and G does not contain cycles of length 4, then G admits an acyclic coloring with O(Δ(k-1)/(k-2)) colors such that no path with k vertices is bichromatic.
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Hou, J., Wu, S. Acyclic coloring of graphs without bichromatic long path. Front. Math. China 10, 1343–1354 (2015). https://doi.org/10.1007/s11464-015-0497-4
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DOI: https://doi.org/10.1007/s11464-015-0497-4