## Site.VertexPartition History

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June 15, 2016, at 01:59 PM
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- Changed line 46 from:

|| ||(I,F5) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F5) ||F. Dross, M. Montassier, A. Pinlou. '''Partitioning sparse graphs into an independent set and a forest of bounded degree'''. http://arxiv.org/pdf/1606.04394v1.pdf

Changed line 50 from:

|| ||(I,F3) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F3) ||F. Dross, M. Montassier, A. Pinlou. '''Partitioning sparse graphs into an independent set and a forest of bounded degree'''. http://arxiv.org/pdf/1606.04394v1.pdf

Changed line 53 from:

|| ||(I,F2) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F2) ||F. Dross, M. Montassier, A. Pinlou. '''Partitioning sparse graphs into an independent set and a forest of bounded degree'''. http://arxiv.org/pdf/1606.04394v1.pdf

June 15, 2016, at 01:57 PM
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- Changed line 46 from:

|| ||(I,F5) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F5) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

Changed line 50 from:

|| ||(I,F3) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F3) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

Changed line 53 from:

|| ||(I,F2) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

to:

|| ||(I,F2) ||F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

June 15, 2016, at 01:53 PM
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- Added line 46:

|| ||(I,F5) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

Added line 50:

|| ||(I,F3) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

Added line 53:

|| ||(I,F2) || F. Dross, M. Montassier, A. Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree.

January 11, 2016, at 10:39 AM
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- Changed lines 33-34 from:

|| ||(F,F5)||François Dross, Mickael Montassier, Alexandre Pinlou. '''Partitioning a triangle-free planar graph into a forest and a forest of bounded degree'''. http://arxiv.org/abs/1601.01523, 2016.

to:

|| ||(F,F5)||François Dross, Mickael Montassier, Alexandre Pinlou. '''Partitioning a triangle-free planar graph into a forest and a forest of bounded degree'''. http://arxiv.org/pdf/1601.01523v1.pdf

January 11, 2016, at 10:38 AM
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- Added lines 33-34:

|| ||(F,F5)||François Dross, Mickael Montassier, Alexandre Pinlou. '''Partitioning a triangle-free planar graph into a forest and a forest of bounded degree'''. http://arxiv.org/abs/1601.01523, 2016.

December 02, 2015, at 02:45 PM
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- Changed line 35 from:

|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh. '''(1,k)-coloring of graphs with girth at least 5 on a surface'''. arXiv:1412.0344, 2014.

to:

|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh. '''(1,k)-coloring of graphs with girth at least 5 on a surface'''. http://arxiv.org/pdf/1412.0344v1.pdf

Changed line 43 from:

|| ||(O12,O12)||L. Esperet and P. Ochem.'''Islands in graphs on surfaces'''. http://arxiv.org/pdf/1402.2475v3.pdf

to:

|| ||(O12,O12)||L. Esperet and P. Ochem. '''Islands in graphs on surfaces'''. http://arxiv.org/pdf/1402.2475v3.pdf

December 02, 2015, at 02:43 PM
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- Changed lines 13-14 from:

* dk the class of k-degenerate graphs

to:

* dk the class of k-degenerate graphs,

* Ok the class of graphs of order k.

* Ok the class of graphs of order k.

Added line 43:

|| ||(O12,O12)||L. Esperet and P. Ochem.'''Islands in graphs on surfaces'''. http://arxiv.org/pdf/1402.2475v3.pdf

December 02, 2015, at 08:45 AM
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- Changed line 55 from:

Thanks to Ilkyoo Choi, Pascal Ochem.

to:

Thanks to Ilkyoo Choi, François, Dross, Pascal Ochem.

December 02, 2015, at 08:37 AM
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- Changed line 41 from:

|| ||(P14,P14)|| M. Axenovich, T. Ueckerdt, and P. Weiner. '''Splitting planar graphs of girth 6 into two linear forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

to:

|| ||(P14,P14)||M. Axenovich, T. Ueckerdt, and P. Weiner. '''Splitting planar graphs of girth 6 into two linear forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

Changed line 55 from:

Thanks to Pascal Ochem.

to:

Thanks to Ilkyoo Choi, Pascal Ochem.

December 02, 2015, at 08:36 AM
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- Changed lines 41-42 from:

|| ||(P14,P14)|| M. Axenovich, T. Ueckerdt, and P. Weiner. '''Splitting planar graphs of girth 6 into two linear

forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

to:

|| ||(P14,P14)|| M. Axenovich, T. Ueckerdt, and P. Weiner. '''Splitting planar graphs of girth 6 into two linear forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

December 02, 2015, at 08:36 AM
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- Added lines 8-9:

* P the class of paths,

* Pk the class of paths ol length at most k,

* Pk the class of paths ol length at most k,

Added lines 41-42:

forests with short paths'''. http://arxiv.org/pdf/1507.02815.pdf

July 28, 2015, at 01:38 PM
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- Changed line 52 from:

Thank you to Pascal Ochem.

to:

Thanks to Pascal Ochem.

June 12, 2015, at 01:53 PM
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- Changed line 52 from:

Thank you to Pascal Ochem

to:

Thank you to Pascal Ochem.

June 12, 2015, at 01:52 PM
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.

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June 12, 2015, at 01:52 PM
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!!! Acknowledgement

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!!! Acknowledgements

June 12, 2015, at 01:51 PM
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!!! Acknowledgment

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!!! Acknowledgement

June 12, 2015, at 01:51 PM
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----

!!! Acknowledgment

Thank you to Pascal Ochem

!!! Acknowledgment

Thank you to Pascal Ochem

June 12, 2015, at 01:44 PM
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- Deleted line 39:

|| ||(S2,S2) ||O.V. Borodin and A.O. Ivanova. '''List strong linear 2-arboricity of sparse graphs'''. Journal of Graph Theory, 67(2):83-90, 2011.

Changed line 41 from:

|| ||{-(I,D2)-} ||M. Montassier and P. Ochem. '''Near-colorings: non-colorable graphs and NP-completness'''. http://arxiv.org/abs/1306.0752

to:

|| ||{-(I,D2)-} ||M. Montassier and P. Ochem. '''Near-colorings: non-colorable graphs and NP-completness'''. Electronic Journal of Combinatorics,Volume 22, Issue 1, 2015.

June 10, 2015, at 06:07 PM
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||10 ||(I,D1) ?|| Open question.

to:

||10 ||(I,D1) ?||Open question.

June 10, 2015, at 06:06 PM
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- Added line 45:

||10 ||(I,D1) ?|| Open question.

June 10, 2015, at 06:05 PM
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- Changed line 37 from:

|| ||{-(I,Fd)-} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

|| ||{-(I,Dk)-} for any k ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 06:02 PM
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- Changed line 13 from:

Observe that: d1=F, D0=I, D1=S1

to:

Observe that: d1=F, D0=I, D1=S1=F1

June 10, 2015, at 06:01 PM
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- Added line 44:

||9 ||{-(I,D1)-} ||L. Esperet, M. Montassier, P. Ochem, and A. Pinlou. '''A complexity dichotomy for the coloring of sparse graphs'''. Journal of Graph Theory, 73(1):85-102, 2013.

June 10, 2015, at 05:57 PM
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- Added line 42:

|| ||{-(I,D2)-} ||M. Montassier and P. Ochem. '''Near-colorings: non-colorable graphs and NP-completness'''. http://arxiv.org/abs/1306.0752

June 10, 2015, at 05:41 PM
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- Changed line 37 from:

|| ||no {-(I,Fd)-} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

|| ||{-(I,Fd)-} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 05:41 PM
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- Changed line 37 from:

|| ||no {(I,Fd)} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

|| ||no {-(I,Fd)-} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 05:40 PM
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- Changed line 37 from:

|| ||no (I,Fd) for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

|| ||no {(I,Fd)} for any d ||O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 05:39 PM
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- Changed line 37 from:

||no (I,Fd) for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

|| ||no (I,Fd) for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 05:39 PM
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- Changed line 37 from:

||{(I,Fd)} for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

to:

||no (I,Fd) for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 10, 2015, at 05:38 PM
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- Added line 37:

||{(I,Fd)} for any d || O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. '''Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k'''. Journal of Graph Theory, 65(2):83-93, 2010.

June 09, 2015, at 09:39 PM
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- Added lines 14-16:

Consider i classes of graphs G1, . . . , Gi. A (G1, . . . , Gi)-partition of a graph G is a vertex-partition into i sets V1,...,Vi such that, for all 1 <= j <= i, the graph G[Vj] induced by Vj belongs to Gj. For example, a (I,F,d2)-partition is a vertex-partition into three sets V1,V2,V3 such that G[V1] is an empty graph, G[V2] is a forest, and G[V3] is a 2-degenerate graph.

June 09, 2015, at 09:36 PM
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- Added lines 14-15:

----

!!! Results

!!! Results

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to:

.

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[=comment .=]

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.

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[=comment .=]

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!!.

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.

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||

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!!.

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- Changed lines 38-39 from:

----

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||

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!!

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----

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!! coucou

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!!

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to:

!! coucou

----

----

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Deleted line 38:

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----

June 09, 2015, at 05:00 PM
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- Added lines 38-39:

Keywords: graph theory, graphs, partition, vertex, planar graphs, given girth

June 09, 2015, at 04:59 PM
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- Changed line 34 from:

|| ||(S2,S2) ||O.V. Borodin and A.O. Ivanova. List strong linear 2-arboricity of sparse graphs. Journal of Graph Theory, 67(2):83-90, 2011.

to:

|| ||(S2,S2) ||O.V. Borodin and A.O. Ivanova. '''List strong linear 2-arboricity of sparse graphs'''. Journal of Graph Theory, 67(2):83-90, 2011.

June 09, 2015, at 04:58 PM
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- Added lines 12-13:

Observe that: d1=F, D0=I, D1=S1

June 09, 2015, at 04:56 PM
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- Changed line 33 from:

|| ||(D1,D1) ||O.V. Borodin, A. Kostochka, M. Yancey. '''On 11-improper 22-coloring of sparse graphs'''. Discrete Mathematics, 313(22):2638-2649, 2013.

to:

|| ||(D1,D1) ||O.V. Borodin, A. Kostochka, M. Yancey. '''On 1-improper 2-coloring of sparse graphs'''. Discrete Mathematics, 313(22):2638-2649, 2013.

June 09, 2015, at 04:56 PM
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- Added line 33:

|| ||(D1,D1) ||O.V. Borodin, A. Kostochka, M. Yancey. '''On 11-improper 22-coloring of sparse graphs'''. Discrete Mathematics, 313(22):2638-2649, 2013.

Deleted line 34:

|| ||(D1,D1) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

June 09, 2015, at 11:18 AM
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- June 09, 2015, at 11:14 AM
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- Changed lines 35-36 from:

||9 || ||

||10 || ||

||10 || ||

to:

||11 ||(I,D1) ||J. Kim, A. Kostochka, X. Zhu.'''Improper coloring of sparse graphs with a given girth, I: (0,1)-colorings of triangle-free graphs'''. European Journal of Combinatorics, 42:26-48, 2014.

June 09, 2015, at 11:09 AM
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- Added lines 8-9:

* S the class of star forests,

* Sk the class of star forests with maximum degree at most k,

* Sk the class of star forests with maximum degree at most k,

Changed lines 31-32 from:

||7 || ||

||8 || ||

||8 || ||

to:

||7 ||(I,D4) ||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.

|| ||(S2,S2) ||O.V. Borodin and A.O. Ivanova. List strong linear 2-arboricity of sparse graphs. Journal of Graph Theory, 67(2):83-90, 2011.

||8 ||(I,D2) ||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.

|| ||(D1,D1) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

|| ||(S2,S2) ||O.V. Borodin and A.O. Ivanova. List strong linear 2-arboricity of sparse graphs. Journal of Graph Theory, 67(2):83-90, 2011.

||8 ||(I,D2) ||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.

|| ||(D1,D1) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

June 09, 2015, at 11:02 AM
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|| ||(D2,D2) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

to:

|| ||(D2,D2) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

||7 || ||

||8 || ||

||9 || ||

||10 || ||

||7 || ||

||8 || ||

||9 || ||

||10 || ||

June 09, 2015, at 11:01 AM
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|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh.'''(1,k)-coloring of graphs with girth at least 5 on a surface'''. arXiv:1412.0344, 2014.

to:

|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh. '''(1,k)-coloring of graphs with girth at least 5 on a surface'''. arXiv:1412.0344, 2014.

Added lines 27-28:

||6 ||(D1,D4) ||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.

|| ||(D2,D2) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

|| ||(D2,D2) ||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

June 09, 2015, at 10:59 AM
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|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh.'''(1,k)-coloring of graphs with girth at least 5 on a surface

'''. arXiv:1412.0344, 2014.

'''. arXiv:1412.0344, 2014.

to:

|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh.'''(1,k)-coloring of graphs with girth at least 5 on a surface'''. arXiv:1412.0344, 2014.

June 09, 2015, at 10:59 AM
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|| ||(D1,D10)|| .'''(1,k)-coloring of graphs with girth at least 5 on a surface

to:

|| ||(D1,D10)||H. Choi, I. Choi, J. Jeong, G. Suh.'''(1,k)-coloring of graphs with girth at least 5 on a surface

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|| ||(D3,D5)|| I. Choi and A. Raspaud. '''Planar graphs with girth at least 5 are (3,5)-colorable'''. Discrete Mathematics, 318(4):661–667, 2015.

to:

|| ||(D3,D5)||I. Choi and A. Raspaud. '''Planar graphs with girth at least 5 are (3,5)-colorable'''. Discrete Mathematics, 318(4):661–667, 2015.

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|| ||(D1,D10)||

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|| ||(D1,D10)|| .'''(1,k)-coloring of graphs with girth at least 5 on a surface

'''. arXiv:1412.0344, 2014.

'''. arXiv:1412.0344, 2014.

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|| ||(D3,D5)||

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|| ||(D3,D5)|| I. Choi and A. Raspaud. '''Planar graphs with girth at least 5 are (3,5)-colorable'''. Discrete Mathematics, 318(4):661–667, 2015.

June 09, 2015, at 10:46 AM
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|| ||(D4,D4)||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree''', Journal of Graph Theory 52(3):181-199, 2006.

to:

|| ||(D4,D4)||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree'''. Journal of Graph Theory, 52(3):181-199, 2006.

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|| ||(D4,D4)||

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|| ||(D4,D4)||F. Havet, J.-S. Sereni. '''Improper choosability of graphs and maximum average degree''', Journal of Graph Theory 52(3):181-199, 2006.

June 09, 2015, at 10:42 AM
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|| ||(D2,D6)||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory Series B, 104:72-80, 2014.

to:

|| ||(D2,D6)||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory, Series B, 104:72-80, 2014.

June 09, 2015, at 10:42 AM
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|| ||(I,d2)||Folklore

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|| ||(I,d2)||Folklore

||5||(I,F)||O.V. Borodin, A.N. Glebov. '''On the partition of a planar graph of girth 5 into an empty and an acyclic subgraph''' (russian). Diskretnyi Analiz i Issledovanie Operatsii, 8(4):34–53, 2001.

|| ||(D1,D10)||

|| ||(D2,D6)||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory Series B, 104:72-80, 2014.

|| ||(D3,D5)||

|| ||(D4,D4)||

||5||(I,F)||O.V. Borodin, A.N. Glebov. '''On the partition of a planar graph of girth 5 into an empty and an acyclic subgraph''' (russian). Diskretnyi Analiz i Issledovanie Operatsii, 8(4):34–53, 2001.

|| ||(D1,D10)||

|| ||(D2,D6)||O.V. Borodin, A.V. Kostochka. '''Defective 2-colorings of sparse graphs'''. Journal of Combinatorial Theory Series B, 104:72-80, 2014.

|| ||(D3,D5)||

|| ||(D4,D4)||

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|| ||(F,F)|| Folklore

|| ||(I,d2)|| Folklore

|| ||(I,d2)|| Folklore

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|| ||(F,F)||Folklore

|| ||(I,d2)||Folklore

|| ||(I,d2)||Folklore

June 09, 2015, at 10:36 AM
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||4 ||(I,I,I) ||H. Grötzsch. '''Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel''', Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120, 1959.

to:

||4 ||(I,I,I) ||H. Grötzsch. '''Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel''', Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120, 1959.

|| ||(F,F)|| Folklore

|| ||(I,d2)|| Folklore

|| ||(F,F)|| Folklore

|| ||(I,d2)|| Folklore

June 09, 2015, at 10:35 AM
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||4 || ||

to:

||4 ||(I,I,I) ||H. Grötzsch. '''Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel''', Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120, 1959.

June 09, 2015, at 10:33 AM
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- Changed lines 13-18 from:

||3 ||(I,I,I,I) ||Appel K., Haken W. '''Every Planar Map is Four Colorable Part I'''. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| ||(I,F,F)||

|| ||(F2,F2,F2)||

|| ||(F,d2)||

|| ||(I,d3)||

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| ||(I,F,F)||

|| ||(F2,F2,F2)||

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to:

||3 ||(I,I,I,I) ||K. Appel K, W. Haken. '''Every Planar Map is Four Colorable Part I'''. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||K. Appel, W. Haken W, J. Koch. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| ||(I,F,F)||O.V. Borodin. '''A proof of Grünbaum’s conjecture on the acyclic 5-colorability of planar graphs''' (russian). Dokl. Akad. Nauk SSSR, 231(1):18–20, 1976.

|| ||(F2,F2,F2)||K.S. Poh. '''On the linear vertex-arboricity of a plane graph'''. Journal of Graph Theory, 14(1):73–75, 1990.

|| ||(F,d2)||C. Thomassen. '''Decomposing a planar graph into degenerate graphs'''. Journal of Combinatorial Theory, Series B, 65(2):305–314, 1995.

|| ||(I,d3)||C. Thomassen. '''Decomposing a planar graph into an independent set and a 3-degenerate graph'''. Journal of Combinatorial Theory, Series B, 83(2):262–271, 2001.

|| || ||K. Appel, W. Haken W, J. Koch. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| ||(I,F,F)||O.V. Borodin. '''A proof of Grünbaum’s conjecture on the acyclic 5-colorability of planar graphs''' (russian). Dokl. Akad. Nauk SSSR, 231(1):18–20, 1976.

|| ||(F2,F2,F2)||K.S. Poh. '''On the linear vertex-arboricity of a plane graph'''. Journal of Graph Theory, 14(1):73–75, 1990.

|| ||(F,d2)||C. Thomassen. '''Decomposing a planar graph into degenerate graphs'''. Journal of Combinatorial Theory, Series B, 65(2):305–314, 1995.

|| ||(I,d3)||C. Thomassen. '''Decomposing a planar graph into an independent set and a 3-degenerate graph'''. Journal of Combinatorial Theory, Series B, 83(2):262–271, 2001.

June 09, 2015, at 10:28 AM
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||4 || ||

||4 || ||

June 09, 2015, at 10:28 AM
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- Changed lines 14-18 from:

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

to:

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| ||(I,F,F)||

|| ||(F2,F2,F,2)||

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June 09, 2015, at 10:26 AM
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||3 ||(I,I,I,I) ||Appel K., Haken W. Every Planar Map is Four Colorable Part I. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. '''Every Planar Map is Four Colorable Part I'''. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.

June 09, 2015, at 10:26 AM
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- Changed lines 13-14 from:

||3 ||(I,I,I,I) ||Appel K., Haken W. '''Every Planar Map is Four Colorable Part I. Discharging"', Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J. "'Every Planar Map is Four Colorable Part II. Reducibility"', Illinois Journal of Mathematics 21: 491–567, 1977.

|| || ||Appel K., Haken W., Koch J. "'Every Planar Map is Four Colorable Part II. Reducibility"', Illinois Journal of Mathematics 21: 491–567, 1977.

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. Every Planar Map is Four Colorable Part I. Discharging, Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

|| || ||Appel K., Haken W., Koch J. '''Every Planar Map is Four Colorable Part II'''. Reducibility, Illinois Journal of Mathematics 21: 491–567, 1977.

June 09, 2015, at 10:25 AM
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- Changed lines 13-14 from:

||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

|| || ||Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. '''Every Planar Map is Four Colorable Part I. Discharging"', Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J. "'Every Planar Map is Four Colorable Part II. Reducibility"', Illinois Journal of Mathematics 21: 491–567, 1977.

|| || ||Appel K., Haken W., Koch J. "'Every Planar Map is Four Colorable Part II. Reducibility"', Illinois Journal of Mathematics 21: 491–567, 1977.

June 09, 2015, at 10:25 AM
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- Changed lines 13-14 from:

||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977. \\ Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977.

|| || ||Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

|| || ||Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

June 09, 2015, at 10:24 AM
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||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977. Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977. \\ Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

June 09, 2015, at 10:24 AM
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|| border=0 align=left width=100%

to:

|| border=0 align=left

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||3 ||(I,I,I,I) ||Appel

to:

||3 ||(I,I,I,I) ||Appel K., Haken W. "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490, 1977. Appel K., Haken W., Koch J., "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567, 1977.

June 09, 2015, at 10:21 AM
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- Changed lines 11-12 from:

|| border=0 width=100%

|| Girth || Vertex-partitions || References

|| Girth || Vertex-partitions || References

to:

|| border=0 align=left width=100%

||Girth ||Vertex-partitions ||References

||Girth ||Vertex-partitions ||References

June 09, 2015, at 10:20 AM
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||Girth ||Vertex-partitions ||References

to:

|| Girth || Vertex-partitions || References

June 09, 2015, at 10:20 AM
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(:table border=0 width=100%:)

(:cell:)

Girth

(:cell:)

Vertex-partitions

(:cell:)

References

(:cellnr:)

3

(:cell:)

(I,I,I,I)

(:cell:)

Appel,

(:tableend:)

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Girth

(:cell:)

Vertex-partitions

(:cell:)

References

(:cellnr:)

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(:tableend:)

to:

|| border=0 width=100%

||Girth ||Vertex-partitions ||References

||3 ||(I,I,I,I) ||Appel

||Girth ||Vertex-partitions ||References

||3 ||(I,I,I,I) ||Appel

June 09, 2015, at 10:18 AM
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(:table border=1 width=100%:)

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(:table border=0 width=100%:)

June 09, 2015, at 10:17 AM
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3

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(I,I,I,I)

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to:

Appel,

June 09, 2015, at 10:16 AM
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Girth

to:

Girth

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Vertex-partition

to:

Vertex-partitions

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Reference

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References

June 09, 2015, at 10:16 AM
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Girth

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Vertex-partition

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Reference

June 09, 2015, at 10:15 AM
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- Changed lines 9-24 from:

* dk the class of k-degenerate graphs

to:

* dk the class of k-degenerate graphs

(:table border=1 width=100%:)

(:cell:)

(:cell:)

(:cell:)

(:cellnr:)

(:cell:)

(:cell:)

(:tableend:)

(:table border=1 width=100%:)

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June 09, 2015, at 10:14 AM
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* Dk the class of graphs with mximum degree at most k,

to:

* Dk the class of graphs with maximum degree at most k,

June 09, 2015, at 10:14 AM
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- Changed lines 5-9 from:

* (:latex:) $\cal I$

\begin{equation}

\prod_{n=2}^N\frac{n}{n-1} = N

\end{equation}

(:latexend:)

\begin{equation}

\prod_{n=2}^N\frac{n}{n-1} = N

\end{equation}

(:latexend:)

to:

* I the class of empty graphs,

* F the class of forests,

* Fk the class of forests with maximum degree at most k,

* Dk the class of graphs with mximum degree at most k,

* dk the class of k-degenerate graphs

* F the class of forests,

* Fk the class of forests with maximum degree at most k,

* Dk the class of graphs with mximum degree at most k,

* dk the class of k-degenerate graphs

June 09, 2015, at 10:10 AM
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- Changed lines 5-9 from:

* $\cal I$

to:

* (:latex:) $\cal I$

\begin{equation}

\prod_{n=2}^N\frac{n}{n-1} = N

\end{equation}

(:latexend:)

\begin{equation}

\prod_{n=2}^N\frac{n}{n-1} = N

\end{equation}

(:latexend:)

June 09, 2015, at 10:06 AM
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----

to:

----

Let us consider the following classes of graphs:

* $\cal I$

Let us consider the following classes of graphs:

* $\cal I$

June 09, 2015, at 10:06 AM
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- June 09, 2015, at 09:54 AM
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- Changed line 1 from:

!!! Vertex-partitions of planar graphs

to:

!! Vertex-partitions of planar graphs

June 09, 2015, at 09:54 AM
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- Added lines 1-2:

!!! Vertex-partitions of planar graphs

----

----