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2008 | 28 | 2 | 323-333
Tytuł artykułu

On acyclic colorings of direct products

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
28
Numer
2
Strony
323-333
Opis fizyczny
Daty
wydano
2008
otrzymano
2007-12-13
poprawiono
2008-02-18
zaakceptowano
2008-02-20
Twórcy
  • University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia
  • University of Maribor, FERI, Smetanova 17, 2000 Maribor, Slovenia
Bibliografia
  • [1] N. Alon, C. McDiarmid and B. Reed, Acyclic colouring of graphs, Random Structures and Algorithms 2 (1991) 277-288, doi: 10.1002/rsa.3240020303.
  • [2] N. Alon, B. Mohar and D. P. Sanders, On acyclic colorings of graphs on surfaces, Israel J. Math. 94 (1996) 273-283, doi: 10.1007/BF02762708.
  • [3] O.V. Borodin, On decomposition of graphs into degenerate subgraphs, Diskretny Analys, Novosibirsk 28 (1976) 3-12 (in Russian).
  • [4] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236, doi: 10.1016/0012-365X(79)90077-3.
  • [5] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973) 390-412, doi: 10.1007/BF02764716.
  • [6] D. Duffus, B. Sands and R.E. Woodrow, On the chromatic number of the product of graphs, J. Graph Theory 9 (1985) 487-495, doi: 10.1002/jgt.3190090409.
  • [7] M. El-Zahar and N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985) 121-126, doi: 10.1007/BF02579374.
  • [8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, New York, 2000).
  • [9] R.E. Jamison and G.L. Matthews, Acyclic colorings of products of cycles, manuscript 2005.
  • [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.
  • [11] B. Mohar, Acyclic colorings of locally planar graphs, European J. Combin. 26 (2005) 491-503, doi: 10.1016/j.ejc.2003.12.016.
  • [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.
  • [13] X. Zhu, A survey on Hedetniemi's conjecture, Taiwanese J. Math. 2 (1998) 1-24.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1408
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