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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1042,
author = {Piotr Borowiecki and Mariusz Ha\l uszczak},
title = {Partitions of some planar graphs into two linear forests},
journal = {Discussiones Mathematicae Graph Theory},
volume = {17},
year = {1997},
pages = {95-102},
zbl = {0905.05061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1042}
}
Piotr Borowiecki; Mariusz Hałuszczak. Partitions of some planar graphs into two linear forests. Discussiones Mathematicae Graph Theory, Tome 17 (1997) pp. 95-102. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1042/
[000] [1] M. Borowiecki, P. Mihók, Hereditary properties of graphs, in: Advances in Graph Theory (Vishwa International Publications, 1991) 41-68.
[001] [2] P. Borowiecki, P-Bipartitions of Graphs, Vishwa International J. GraphTheory 2 (1993) 109-116.
[002] [3] I. Broere, C.M. Mynhardt, Generalized colourings of outerplanar and planar graphs, in: Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 151-161.
[003] [4] W. Goddard, Acyclic colourings of planar graphs, Discrete Math. 91 (1991) 91-94, doi: 10.1016/0012-365X(91)90166-Y. | Zbl 0742.05041
[004] [5] T.R. Jensen and B. Toft, Graph Colouring Problems (Wiley-Interscience Publications, New York, 1995). | Zbl 0971.05046
[005] [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.
[006] [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. | Zbl 0705.05016
[007] [8] J. Wang, On point-linear arboricity of planar graphs, Discrete Math. 72 (1988) 381-384, doi: 10.1016/0012-365X(88)90229-4. | Zbl 0665.05010