On acyclic colorings of direct products

Simon Špacapan; Aleksandra Tepeh Horvat

Discussiones Mathematicae Graph Theory (2008)

  • Volume: 28, Issue: 2, page 323-333
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Simon Špacapan, and Aleksandra Tepeh Horvat. "On acyclic colorings of direct products." Discussiones Mathematicae Graph Theory 28.2 (2008): 323-333. <http://eudml.org/doc/270706>.

@article{SimonŠpacapan2008,
abstract = {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.},
author = {Simon Špacapan, Aleksandra Tepeh Horvat},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {coloring; acyclic coloring; distance-two coloring; direct product},
language = {eng},
number = {2},
pages = {323-333},
title = {On acyclic colorings of direct products},
url = {http://eudml.org/doc/270706},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Simon Špacapan
AU - Aleksandra Tepeh Horvat
TI - On acyclic colorings of direct products
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 2
SP - 323
EP - 333
AB - 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.
LA - eng
KW - coloring; acyclic coloring; distance-two coloring; direct product
UR - http://eudml.org/doc/270706
ER -

References

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  8. [8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, New York, 2000). 
  9. [9] R.E. Jamison and G.L. Matthews, Acyclic colorings of products of cycles, manuscript 2005. Zbl1181.05044
  10. [10] R.E. Jamison, G.L. Matthews and J. Villalpando, Acyclic colorings of products of trees, Inform. Process. Lett. 99 (2006) 7-12, doi: 10.1016/j.ipl.2005.11.023. Zbl1184.05043
  11. [11] B. Mohar, Acyclic colorings of locally planar graphs, European J. Combin. 26 (2005) 491-503, doi: 10.1016/j.ejc.2003.12.016. Zbl1058.05024
  12. [12] C. Tardif, The fractional chromatic number of the categorical product of graphs, Combinatorica 25 (2005) 625-632, doi: 10.1007/s00493-005-0038-y. Zbl1101.05035
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