Discussiones Mathematicae Graph Theory 29(2) (2009) 219-239
doi: 10.7151/dmgt.1443

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Mieczysław Borowiecki, Anna Fiedorowicz and Mariusz Hałuszczak

Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Z. Szafrana 4a, Zielona Góra, Poland



For a given graph G and a sequence P1, P2,…, Pn of additive hereditary classes of graphs we define an acyclic (P1, P2,…,Pn)-colouring of G as a partition (V1, V2,…,Vn) of the set V(G) of vertices which satisfies the following two conditions:

  1. G[Vi] ∈ Pi for i = 1,…,n,
  2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that u ∈ Vi and v ∈ Vj is acyclic.

A class R = P1P2 Pn is defined as the set of the graphs having an acyclic (P1, P2,…,Pn)-colouring. If PR, then we say that R is an acyclic reducible bound for P.

In this paper we present acyclic reducible bounds for the class of outerplanar graphs.

Keywords: graph, acyclic colouring, additive hereditary class, outerplanar graph.

2000 Mathematics Subject Classification: 05C75, 05C15, 05C35.


[1] P. Boiron, E. Sopena and L. Vignal, Acyclic improper colorings of graphs, J. Graph Theory 32 (1999) 97-107, doi: 10.1002/(SICI)1097-0118(199909)32:1<97::AID-JGT9>3.0.CO;2-O.
[2] P. Boiron, E. Sopena and L. Vignal, Acyclic improper colourings of graphs with bounded degree, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 49 (1999) 1-9.
[3] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
[4] M. Borowiecki and A. Fiedorowicz, On partitions of hereditary properties of graphs, Discuss. Math. Graph Theory 26 (2006) 377-387, doi: 10.7151/dmgt.1330.
[5] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236, doi: 10.1016/0012-365X(79)90077-3.
[6] O.V. Borodin, A.V. Kostochka and D.R. Woodall, Acyclic colorings of planar graphs with large girth, J. London Math. Soc. 60 (1999) 344-352, doi: 10.1112/S0024610799007942.
[7] M.I. Burstein, Every 4-valent graph has an acyclic 5-coloring, Soobsc. Akad. Nauk Gruzin SSR 93 (1979) 21-24 (in Russian).
[8] R. Diestel, Graph Theory (Springer, Berlin, 1997).
[9] B. Grunbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973) 390-412, doi: 10.1007/BF02764716.
[10] D.B. West, Introduction to Graph Theory, 2nd ed. (Prentice Hall, Upper Saddle River, 2001).

Received 13 December 2007
Revised 4 July 2008
Accepted 23 October 2008