Equitable partition of plane graphs with independent crossings into induced forests

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into $m$ induced forests for any $m\geq 8$. Moreover, we decrease this lower bound 8 for $m$ to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.

[1]  Xin Zhang,et al.  A conjecture on equitable vertex arboricity of graphs , 2012, ArXiv.

[2]  Xin Zhang.  Drawing complete multipartite graphs on the plane with restrictions on crossings , 2013, ArXiv.

[3]  Guanghui Wang,et al.  Equitable vertex arboricity of 5-degenerate graphs , 2017, J. Comb. Optim..

[4]  Oleg V. Borodin.  A new proof of the 6 color theorem , 1995, J. Graph Theory.

[5]  G. Ringel.  Ein Sechsfarbenproblem auf der Kugel , 1965 .

[6]  D. Král,et al.  Coloring plane graphs with independent crossings , 2010 .

[7]  Giuseppe Liotta,et al.  An annotated bibliography on 1-planarity , 2017, Comput. Sci. Rev..

[8]  Xin Zhang.  Equitable vertex arboricity of subcubic graphs , 2016, Discret. Math..

[9]  Xin Zhang,et al.  Equitable vertex arboricity of graphs , 2013, Discret. Math..

[10]  Svante Janson,et al.  The Chromatic Number , 2011 .

[11]  Xin Zhang,et al.  Equitable partition of graphs into induced linear forests , 2020, J. Comb. Optim..

[12]  Yan Li,et al.  Equitable vertex arboricity of $d$-degenerate graphs , 2019, ArXiv.

[13]  Bi Li,et al.  Tree-coloring problems of bounded treewidth graphs , 2019, J. Comb. Optim..

[14]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[15]  Louis Esperet,et al.  Equitable partition of graphs into induced forests , 2015, Discret. Math..

[16]  André Raspaud,et al.  Acyclic colouring of 1-planar graphs , 2001, Discret. Appl. Math..

[17]  Hal A. Kierstead,et al.  A note on relaxed equitable coloring of graphs , 2011, Inf. Process. Lett..

[18]  P. Lax.  Proof of a conjecture of P. Erdös on the derivative of a polynomial , 1944 .

[19]  Xin Zhang.  Equitable vertex arboricity of planar graphs , 2014, ArXiv.