Geodesic
Decomposing a~planar graph into a~forest and a~subgraph of restricted maximum degree
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 296-299.

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We disprove the conjecture of He, Hou, Lih, Shao, Wang and Zhu that every plane graph $G$ can be edge-partitioned into a forest and a subgraph of the maximum degree at most $\lceil\Delta(G)/2\rceil+1$. 
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O. V. Borodin; A. O. Ivanova; B. S. Stechkin. Decomposing a~planar graph into a~forest and a~subgraph of restricted maximum degree. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 296-299. http://geodesic.mathdoc.fr/item/SEMR_2007_4_a18/

[1] 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 | MR | Zbl

[2] 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

[3] O. V. Borodin, A. V. Kostochka, N. N. Sheikh, and Gexin Yu, “Decomposing a planar graph with girth nine into a forest and a matching”, Europ. J. Combin. (to appear)

[4] O. V. Borodin, A. V. Kostochka, N. N. Sheikh, and Gexin Yu, $M$-degrees of $C_4$-free planar graphs, submitted