Geodesic
The List Edge Coloring and List Total Coloring of Planar Graphs with Maximum Degree at Least 7
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1005-1024.

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

A graph G is edge k-choosable (respectively, total k-choosable) if, whenever we are given a list L(x) of colors with |L(x)| = k for each x ∈ E(G) (x ∈ E(G) ∪ V (G)), we can choose a color from L(x) for each element x such that no two adjacent (or incident) elements receive the same color. The list edge chromatic index χ_l^′(G) (respectively, the list total chromatic number χ_l^′′(G)) of G is the smallest integer k such that G is edge (respectively, total) k-choosable. In this paper, we focus on a planar graph G, with maximum degree Δ (G) ≥ 7 and with some structural restrictions, satisfies χ_l^′(G) = Δ (G) and χ_l^′′(G) = Δ (G) + 1.
Keywords: planar graph, list edge coloring, list total coloring 
@article{DMGT_2020_40_4_a4,
     author = {Sun, Lin and Wu, Jianliang and Wang, Bing and Liu, Bin},
     title = {The {List} {Edge} {Coloring} and {List} {Total} {Coloring} of {Planar} {Graphs} with {Maximum} {Degree} at {Least} 7},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {1005--1024},
     publisher = {mathdoc},
     volume = {40},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a4/}
}
TY  - JOUR
AU  - Sun, Lin
AU  - Wu, Jianliang
AU  - Wang, Bing
AU  - Liu, Bin
TI  - The List Edge Coloring and List Total Coloring of Planar Graphs with Maximum Degree at Least 7
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2020
SP  - 1005
EP  - 1024
VL  - 40
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a4/
LA  - en
ID  - DMGT_2020_40_4_a4
ER  - 
%0 Journal Article
%A Sun, Lin
%A Wu, Jianliang
%A Wang, Bing
%A Liu, Bin
%T The List Edge Coloring and List Total Coloring of Planar Graphs with Maximum Degree at Least 7
%J Discussiones Mathematicae. Graph Theory
%D 2020
%P 1005-1024
%V 40
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a4/
%G en
%F DMGT_2020_40_4_a4
Sun, Lin; Wu, Jianliang; Wang, Bing; Liu, Bin. The List Edge Coloring and List Total Coloring of Planar Graphs with Maximum Degree at Least 7. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1005-1024. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a4/

[1] N. Alon and M. Tarsi, Colorings and orientation of graphs, Combinatorica 12 (1992) 125–134. doi:10.1007/BF01204715

[2] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999) 7–29. doi:10.1017/S0963548398003411

[3] M. Behzad, Graphs and Their Chromatic Numbers (Doctoral Thesis, Michigan State University, East Lansing, MI, 1965).

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

[5] M. Bonamy, Planar graphs with maximum degree Δ at least 8 are (Δ + 1)-edge-choosable, SIAM J. Discrete Math. 29 (2015) 1735–1763. doi:10.1137/130927449

[6] M. Bonamy, B. Lévêque and A. Pinlou, Planar graphs with Δ ≥ 7 and no triangle adjacent to a C4 are minimally edge and total choosable, Discrete Math. Theoret. Comput. Sci. 17 (2016) 131–146. doi:10.1006/jctb.1997.1780

[7] O.V. Borodin, A.V. Kostochka and D.R. Woodall, List edge and list total colourings of multigraphs, J. Combin. Theory Ser. B 71 (1997) 184–204.

[8] O.V. Borodin, A generalization of Kotzig’s theorem on prescribed edge coloring of planar graphs, Mat. Zametki 48 (1990) 22–28, in Russian.

[9] A.J. Harris, Problems and Conjectures in Extremal Graph Theory, Ph. D. Dissertation (Cambridge, Cambridge University, 1984).

[10] J.F. Hou, G.Z. Liu and J.S. Cai, List edge and list total colorings of planar graphs without 4-cycles, Theoret. Comput. Sci. 369 (2006) 250–255. doi:10.1016/j.tcs.2006.08.043

[11] J.F. Hou, G.Z. Liu and J.L. Wu, Some results on list total colorings of planar graphs, Lecture Notes in Comput. Sci. 4489 (2007) 320–328. doi:10.1007/978-3-540-72588-6_53

[12] T.R. Jensen and B. Toft, Graph Coloring Problems (Wiley Interscience, New York, 1995).

[13] M. Juvan, B. Mohar and R. Škrekovski, Graphs of degree 4 are 5-edge-choosable, J. Graph Theory 32 (1999) 250–262. doi:10.1002/(SICI)1097-0118(199911)32:3〈250::AID-JGT5〉3.0.CO;2-R

[14] M. Juvan, B. Mohar and R. Škrekovski, List total colourings of graphs, Combin. Probab. Comput. 7 (1998) 181–188. doi:10.1017/S0963548397003210

[15] Q.L. Lu, Z.K. Miao and Y.Q. Wang, Sufficient conditions for a planar graph to be list edge Δ-colorable and list totally (Δ + 1)-colorable, Discrete Math. 313 (2013) 575–580. doi:10.1016/j.disc.2012.11.026

[16] V.G. Vizing, Some unsolved problems in graph theory, Uspekhi Mat. Nauk 23 (1968) 117–134, in Russian. doi:10.1070/RM1968v023n06ABEH001252

[17] W.F. Wang and K.W. Lih, Choosability, edge choosability, and total choosability of outerplane graphs, European J. Combin. 22 (2001) 71–78. doi:10.1006/eujc.2000.0430

[18] H.J. Wang, L.D. Wu, X. Zhang, W.L. Wu and B. Liu, A note on the minimum number of choosability of planar graphs, J. Comb. Optim. 31 (2016) 1013–1022. doi:10.1007/s10878-014-9805-2

[19] H.J. Wang, B. Liu, X. Zhang, L.D. Wu, W.L. Wu and H.W. Gao, List edge and list total coloring of planar graphs with maximum degree 8, J. Comb. Optim. 32 (2016) 188–197. doi:10.1007/s10878-015-9870-1

[20] J.L. Wu and P. Wang, List-edge and list-total colorings of graphs embedded on hyperbolic surfaces, Discrete Math. 308 (2008) 6210–6215. doi:10.1016/j.disc.2007.11.044