# Acyclic reducible bounds for outerplanar graphs

• Volume: 29, Issue: 2, page 219-239
• ISSN: 2083-5892

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## Abstract

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For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G\left[{V}_{i}\right]{\in }_{i}$ 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 $u\in {V}_{i}$ and $v\in {V}_{j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring. If ⊆ R, then we say that R is an acyclic reducible bound for . In this paper we present acyclic reducible bounds for the class of outerplanar graphs.

## How to cite

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Mieczysław Borowiecki, Anna Fiedorowicz, and Mariusz Hałuszczak. "Acyclic reducible bounds for outerplanar graphs." Discussiones Mathematicae Graph Theory 29.2 (2009): 219-239. <http://eudml.org/doc/270253>.

@article{MieczysławBorowiecki2009,
abstract = {For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G[V_i] ∈ _i$ 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 $u ∈ V_i$ and $v ∈ V_j$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring. If ⊆ R, then we say that R is an acyclic reducible bound for . In this paper we present acyclic reducible bounds for the class of outerplanar graphs.},
author = {Mieczysław Borowiecki, Anna Fiedorowicz, Mariusz Hałuszczak},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; acyclic colouring; additive hereditary class; outerplanar graph},
language = {eng},
number = {2},
pages = {219-239},
title = {Acyclic reducible bounds for outerplanar graphs},
url = {http://eudml.org/doc/270253},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Mieczysław Borowiecki
AU - Anna Fiedorowicz
AU - Mariusz Hałuszczak
TI - Acyclic reducible bounds for outerplanar graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 2
SP - 219
EP - 239
AB - For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G[V_i] ∈ _i$ 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 $u ∈ V_i$ and $v ∈ V_j$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring. If ⊆ R, then we say that R is an acyclic reducible bound for . In this paper we present acyclic reducible bounds for the class of outerplanar graphs.
LA - eng
KW - graph; acyclic colouring; additive hereditary class; outerplanar graph
UR - http://eudml.org/doc/270253
ER -

## References

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2. [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. Zbl0930.05042
3. [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. Zbl0902.05026
4. [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. Zbl1139.05018
5. [5] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236, doi: 10.1016/0012-365X(79)90077-3. Zbl0406.05031
6. [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. Zbl0940.05032
7. [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. [8] R. Diestel, Graph Theory (Springer, Berlin, 1997).
9. [9] B. Grunbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973) 390-412, doi: 10.1007/BF02764716. Zbl0265.05103
10. [10] D.B. West, Introduction to Graph Theory, 2nd ed. (Prentice Hall, Upper Saddle River, 2001).

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