Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

Affiliation(s)

Department of Information Engineering, Taishan Polytechnic, Tai’an, China.

School of Mathematics, Shandong University, Jinan, China.

Department of Information Engineering, Taishan Polytechnic, Tai’an, China.

School of Mathematics, Shandong University, Jinan, China.

ABSTRACT

It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.

Cite this paper

L. Xue and J. Wu, "Edge Colorings of Planar Graphs without 6-Cycles with Two Chords,"*Open Journal of Discrete Mathematics*, Vol. 3 No. 2, 2013, pp. 83-85. doi: 10.4236/ojdm.2013.32016.

L. Xue and J. Wu, "Edge Colorings of Planar Graphs without 6-Cycles with Two Chords,"

References

[1] S. Fiorini and R. J. Wilson, “Edge-Colorings of Graphs,” In: S. Fiorini and R. J. Wilson, Eds., Edge-Colorings of Graphs, Vol. 16, Pitman, London, 1977.

[2] H. Hind and Y. Zhao, “Edge Colorings of Graphs Embedable in a Surface of Low Genus,” Discrete Mathematics, Vol. 190, No. 1-3, 1998, pp. 107-114. doi:10.1016/S0012-365X(98)00050-8

[3] L. Y. Miao and J. L. Wu, “Edge-Coloring Critical Graphs with High Degree,” Discrete Mathematics, Vol. 257, No. 1, 2002, pp. 169-172. doi:10.1016/S0012-365X(02)00395-3

[4] L. M. Zhang, “Every Planar Graph with Maximum Degree 7 Is of Class 1,” Graphs and Combinatorics, Vol. 16, No. 4, 2000, pp. 467-495. doi:10.1007/s003730070009

[5] D. P. Sanders and Y. Zhao, “Planar Graphs of Maximum Degree Seven Are Class 1,” Journal of Combinatorial Theory, Series B, Vol. 83, No. 2, 2001, pp. 202-212. doi:10.1006/jctb.2001.2047

[6] P. Lam, J. Liu, W. Shiu and J. Wu, “Some Sufficient Conditions for a Planar Graph to Be of Class 1,” Congressus Numerantium, Vol. 136, No. 4, 1999, pp. 201- 205.

[7] G. F. Zhou, “A Note on Graphs of Class 1,” Discrete Mathematics, Vol. 263, No. 1-3, 2003, pp. 339-345. doi:10.1016/S0012-365X(02)00793-8

[8] Y. H. Bu and W. F. Wang, “Some Sufficient Conditions for a Planar Graph of Maximum Degree Six to Be Class 1,” Discrete Mathematics, Vol. 306, No. 13, 2006, pp. 1440-1445. doi:10.1016/j.disc.2006.03.032

[9] W. P. Ni, “Edge Colorings of Planar Graphs with Δ = 6 without Short Cycles Contain Chords,” Journal of Nanjing Normal University, Vol. 34, No. 3, 2011, pp. 19-24 (in Chinese).

[1] S. Fiorini and R. J. Wilson, “Edge-Colorings of Graphs,” In: S. Fiorini and R. J. Wilson, Eds., Edge-Colorings of Graphs, Vol. 16, Pitman, London, 1977.

[2] H. Hind and Y. Zhao, “Edge Colorings of Graphs Embedable in a Surface of Low Genus,” Discrete Mathematics, Vol. 190, No. 1-3, 1998, pp. 107-114. doi:10.1016/S0012-365X(98)00050-8

[3] L. Y. Miao and J. L. Wu, “Edge-Coloring Critical Graphs with High Degree,” Discrete Mathematics, Vol. 257, No. 1, 2002, pp. 169-172. doi:10.1016/S0012-365X(02)00395-3

[4] L. M. Zhang, “Every Planar Graph with Maximum Degree 7 Is of Class 1,” Graphs and Combinatorics, Vol. 16, No. 4, 2000, pp. 467-495. doi:10.1007/s003730070009

[5] D. P. Sanders and Y. Zhao, “Planar Graphs of Maximum Degree Seven Are Class 1,” Journal of Combinatorial Theory, Series B, Vol. 83, No. 2, 2001, pp. 202-212. doi:10.1006/jctb.2001.2047

[6] P. Lam, J. Liu, W. Shiu and J. Wu, “Some Sufficient Conditions for a Planar Graph to Be of Class 1,” Congressus Numerantium, Vol. 136, No. 4, 1999, pp. 201- 205.

[7] G. F. Zhou, “A Note on Graphs of Class 1,” Discrete Mathematics, Vol. 263, No. 1-3, 2003, pp. 339-345. doi:10.1016/S0012-365X(02)00793-8

[8] Y. H. Bu and W. F. Wang, “Some Sufficient Conditions for a Planar Graph of Maximum Degree Six to Be Class 1,” Discrete Mathematics, Vol. 306, No. 13, 2006, pp. 1440-1445. doi:10.1016/j.disc.2006.03.032

[9] W. P. Ni, “Edge Colorings of Planar Graphs with Δ = 6 without Short Cycles Contain Chords,” Journal of Nanjing Normal University, Vol. 34, No. 3, 2011, pp. 19-24 (in Chinese).