Discussiones Mathematicae Graph Theory 29(1) (2009) 87-99
doi: 10.7151/dmgt.1434

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DECOMPOSITIONS OF QUADRANGLE-FREE PLANAR GRAPHS

Oleg V. Borodin

Sobolev Institute of Mathematics
Novosibirsk 630090, Russia
e-mail: brdnoleg@math.nsc.ru

Anna O. Ivanova

Yakutsk State University
Yakutsk, 677000, Russia

e-mail: shmgnanna@mail.ru

Alexandr V. Kostochka

Department of Mathematics
University of Illinois, Urbana, IL 61801, USA
and
Sobolev Institute of Mathematics
Novosibirsk 630090, Russia
e-mail: kostochk@math.uiuc.edu

Naeem N. Sheikh

Department of Mathematics
University of Illinois, Urbana, IL 61801, USA

e-mail: nsheikh@math.uiuc.edu

Abstract

W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.

Keywords and phrases: planar graphs, graph decompositions, quadrangle-free graphs.

2000 Mathematics Subject Classification: 05C15, 05C10, 05C35.

References

[1] A. Bassa, J. Burns, J. Campbell, A. Deshpande, J. Farley, M. Halsey, S. Michalakis, P.-O. Persson, P. Pylyavskyy, L. Rademacher, A. Riehl, M. Rios, J. Samuel, B. Tenner, A. Vijayasaraty, L. Zhao and D. J. Kleitman, Partitioning a planar graph of girth ten into a forest and a matching, manuscript (2004).
[2] O.V. Borodin, Consistent colorings of graphs on the plane, Diskret. Analiz 45 (1987) 21-27 (in Russian).
[3] O. Borodin, A. Kostochka, N. Sheikh and G. Yu, Decomposing a planar graph with girth nine into a forest and a matching, European Journal of Combinatorics 29 (2008) 1235-1241, doi: 10.1016/j.ejc.2007.06.020.
[4] O. Borodin, A. Kostochka, N. Sheikh and G. Yu, M-degrees of quadrangle-free planar graphs, J. Graph Theory 60 (2009) 80-85, doi: 10.1002/jgt.20346.
[5] W. He, X. Hou, K.W. Lih, J. Shao, W. Wang and X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307-317, doi: 10.1002/jgt.10069.
[6] D.J. Kleitman, Partitioning the edges of a girth 6 planar graph into those of a forest and those of a set of disjoint paths and cycles, manuscript.

Received 27 November 2007

Revised 1 August 2008
Accepted 29 August 2008