# Planar graphs with Δ ≥ 8 are (Δ + 1)-edge-choosable

• Marthe Bonamy
Conference paper
Part of the CRM Series book series (PSNS, volume 16)

## Abstract

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is (Δ + 1)-edge-choosable. In 1990, Borodin solved the conjecture for planar graphs with maximum degree Δ ≥ 9, and asked whether the bound could be lowered to 8. We prove here that planar graphs with Δ ≥ 8 are (Δ + 1)-edge-choosable.

## Preview

Unable to display preview. Download preview PDF.

### References

1. [1]
K. Appel and W. Haken, Every map is four colorable: Part 1, Discharging, Illinois J. Math. 21 (1977), 422–490.Google Scholar
2. [2]
Appel, K., W. Haken and J. Koch, Every map is four colorable: Part 2, Reducibility, Illinois J. Math. 21 (1977), pp. 491–567.
3. [3]
M. Bonamy, Planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable, preprint, http://arxiv.org/abs/1303.4025
4. [4]
O. V. Borodin, Generalization of a theorem of Kotzig and a prescribed coloring of the edges of planar graphs, Mathematical Notes 6 (1991), 1186–1190.
5. [5]
O. V. Borodin, Colorings of plane graphs: A survey, Discrete Mathematics 313 (2013), 517–539
6. [6]
O. V. Borodin, A. V. Kostochka and D. R. Woodall, List Edge and List Total Colourings of Multigraphs, Journal of Combinatorial Theory, Series B 71(2) (1997), 184–204.
7. [7]
N. Cohen and F. Havet, Planar graphs with maximum degree Δ ≥ 9 are (Δ + 1)-edge-choosable — A short proof, Discrete Mathematics 310(21) (2010), 3049–3051.
8. [8]
T. Jensen and B. Toft, “Graph Coloring Problems”, Wiley Interscience, New York (1995).Google Scholar
9. [9]
M. Juvan, B. Mohar and R. Škrekovski, Graphs of degree 4 are 5-edge-choosable, Journal of Graph Theory 32 (1999), 250–264.
10. [10]
D. Sanders and Y. Zhao, Planar graphs of maximum degree seven are class I, Journal of Combinatorial Theory, Series B 83(2) (2001), 201–221.
11. [11]
Y. Shen, G. Zheng, W. He and Y. Zhao, Structural properties and edge choosability of planar graphs without 4-cycles, Discrete Mathematics 308 (2008), 5789–5794.
12. [12]
V. G. Vizing, On an estimate of the chromatic class of a p-graph (in russian), Diskret. Analiz 3 (1964), 25–30.
13. [13]
V. G. Vizing,, Critical graphs with given chromatic index (in russian), Diskret. Analiz 5 (1965), 9–17.
14. [14]
V. G. Vizing, Colouring the vertices of a graph with prescribed colours (in russian), Diskret. Analiz 29 (1976), 3–10.
15. [15]
F. W. Wang and K. W. Lih, Choosability and Edge Choosability of Planar Graphs without Five Cycles, Applied Mathematics Letters 15 (2002), 561–565.
16. [16]
L. Zhang and B. Wu, Edge choosability of planar graphs without small cycles, Discrete Mathematics 283 (2004), 289–293.

## Copyright information

© Scuola Normale Superiore Pisa 2013

## Authors and Affiliations

• Marthe Bonamy
• 1
1. 1.LIRMMUniversité Montpellier 2France