Geodesic
On Edge Colorings of 1-Planar Graphs without 5-Cycles with Two Chords
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 301-312.

Voir la notice de l'article provenant de la source Library of Science

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-planar graph with maximum degree ∆ ≥ 8 is edge-colorable with ∆ colors if each of its 5-cycles contains at most one chord.
Keywords: 1-planar graphs, edge coloring, discharging method 
@article{DMGT_2019_39_2_a0,
     author = {Sun, Lin and Wu, Jianliang},
     title = {On {Edge} {Colorings} of {1-Planar} {Graphs} without {5-Cycles} with {Two} {Chords}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {301--312},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a0/}
}
TY  - JOUR
AU  - Sun, Lin
AU  - Wu, Jianliang
TI  - On Edge Colorings of 1-Planar Graphs without 5-Cycles with Two Chords
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2019
SP  - 301
EP  - 312
VL  - 39
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a0/
LA  - en
ID  - DMGT_2019_39_2_a0
ER  - 
%0 Journal Article
%A Sun, Lin
%A Wu, Jianliang
%T On Edge Colorings of 1-Planar Graphs without 5-Cycles with Two Chords
%J Discussiones Mathematicae. Graph Theory
%D 2019
%P 301-312
%V 39
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a0/
%G en
%F DMGT_2019_39_2_a0
Sun, Lin; Wu, Jianliang. On Edge Colorings of 1-Planar Graphs without 5-Cycles with Two Chords. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 301-312. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a0/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan, London, 1976).

[2] Y.H. Bu and W.F. Wang, Some sufficient conditions for a planar graph of maximum degree six to be Class 1, Discrete Math. 306 (2006) 1440–1445. doi:10.1016/j.disc.2006.03.032

[3] P. Erdős and R.J. Wilson, On the chromatic index of almost all graphs, J. Combin. Theory Ser. B 23 (1977) 255–257. doi:10.1016/0095-8956(77)90039-9

[4] R. Luo, L.Y. Miao and Y. Zhao, The size of edge chromatic critical graphs with maximum degree 6, J. Graph Theory 60 (2009) 149–171. doi:10.1002/jgt.20351

[5] S. Fiorini and R.J. Wilson, Edge-colorings of graphs, in: Research Notes in Mathematics 16 (Pitman, London, 1977).

[6] W.P. Ni, Edge colorings of planar graphs with ∆ = 6 without short cycles contain chords, J. Nanjing Norm. Univ. Nat. Sci. Ed. 34 (2011) 19–24.

[7] G. Ringel, Ein Sechsfarbenproblem auf der Kugel, Abh. Math. Semin. Univ. Hambg. 29 (1965) 107–117.

[8] D.P. Sanders and Y. Zhao, Planar graphs of maximum degree seven are Class I, J. Combin. Theory Ser. B 83 (2001) 201–212. doi:10.1006/jctb.2001.2047

[9] V.G. Vizing, Critical graphs with given chromatic class, Diskret. Analiz 5 (1965) 9–17.

[10] V.G. Vizing, Some unsolved problems in graph theory, Russian Math. Surveys 23 (1968) 125–141. doi:10.1070/RM1968v023n06ABEH001252

[11] L. Xue and J.L. Wu, Edge colorings of planar graphs without 6 -cycles with two chords, Open J. Discrete Math. 3 (2013) 83–85. doi:10.4236/ojdm.2013.32016

[12] J.-L. Wu and L. Xue, Edge colorings of planar graphs without 5 -cycles with two chords, Theoret. Comput. Sci. 518 (2014) 124–127. doi:10.1016/j.tcs.2013.07.027

[13] L.M. Zhang, Every planar graph with maximum degree 7 is of Class 1, Graphs Combin. 16 (2000) 467–495. doi:10.1007/s003730070009

[14] X. Zhang, Class two 1 -planar graphs with maximum degree six or seven (2011). arXiv: 1104.4687

[15] X. Zhang and J.-L. Wu, On edge colorings of 1 -planar graphs, Inform. Process. Lett. 111 (2011) 124–128. doi:10.1016/j.ipl.2010.11.001

[16] X. Zhang and G.Z. Liu, On edge colorings of 1 -planar graphs without adjacent triangles, Inform. Process. Lett. 112 (2012) 138–142. doi:10.1016/j.ipl.2011.10.021

[17] X. Zhang and G.Z. Liu, On edge colorings of 1 -planar graphs without chordal 5- cycles, Ars Combin. 104 (2012) 431–436.