# Partitions of some planar graphs into two linear forests

Piotr Borowiecki; Mariusz Hałuszczak

Discussiones Mathematicae Graph Theory (1997)

- Volume: 17, Issue: 1, page 95-102
- ISSN: 2083-5892

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topPiotr Borowiecki, and Mariusz Hałuszczak. "Partitions of some planar graphs into two linear forests." Discussiones Mathematicae Graph Theory 17.1 (1997): 95-102. <http://eudml.org/doc/270623>.

@article{PiotrBorowiecki1997,

abstract = {A linear forest is a forest in which every component is a path. It is known that the set of vertices V(G) of any outerplanar graph G can be partitioned into two disjoint subsets V₁,V₂ such that induced subgraphs ⟨V₁⟩ and ⟨V₂⟩ are linear forests (we say G has an (LF, LF)-partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of the graph G in the plane). We prove that there exists an (LF, LF)-partition for any plane graph G when certain conditions on the degree of the internal vertices and their neighbourhoods are satisfied.},

author = {Piotr Borowiecki, Mariusz Hałuszczak},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {linear forest; bipartition; planar graphs; outerplanar graph; internal vertices; embedding; plane graph},

language = {eng},

number = {1},

pages = {95-102},

title = {Partitions of some planar graphs into two linear forests},

url = {http://eudml.org/doc/270623},

volume = {17},

year = {1997},

}

TY - JOUR

AU - Piotr Borowiecki

AU - Mariusz Hałuszczak

TI - Partitions of some planar graphs into two linear forests

JO - Discussiones Mathematicae Graph Theory

PY - 1997

VL - 17

IS - 1

SP - 95

EP - 102

AB - A linear forest is a forest in which every component is a path. It is known that the set of vertices V(G) of any outerplanar graph G can be partitioned into two disjoint subsets V₁,V₂ such that induced subgraphs ⟨V₁⟩ and ⟨V₂⟩ are linear forests (we say G has an (LF, LF)-partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of the graph G in the plane). We prove that there exists an (LF, LF)-partition for any plane graph G when certain conditions on the degree of the internal vertices and their neighbourhoods are satisfied.

LA - eng

KW - linear forest; bipartition; planar graphs; outerplanar graph; internal vertices; embedding; plane graph

UR - http://eudml.org/doc/270623

ER -

## References

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- [5] T.R. Jensen and B. Toft, Graph Colouring Problems (Wiley-Interscience Publications, New York, 1995). Zbl0971.05046
- [6] P. Mihók, On the vertex partition numbers, in: M. Fiedler, ed., Graphs and Other Combinatorial Topics, Proc. Third Czech. Symp. Graph Theory, Prague, 1982 (Teubner-Verlag, Leipzig, 1983) 183-188.
- [7] K.S. Poh, On the Linear Vertex-Arboricity of a Planar Graph, J. Graph Theory 14 (1990) 73-75, doi: 10.1002/jgt.3190140108. Zbl0705.05016
- [8] J. Wang, On point-linear arboricity of planar graphs, Discrete Math. 72 (1988) 381-384, doi: 10.1016/0012-365X(88)90229-4. Zbl0665.05010

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