- Volume 53 Issue 1
DOI QR Code
TOTAL COLORINGS OF PLANAR GRAPHS WITH MAXIMUM DEGREE AT LEAST 7 AND WITHOUT ADJACENT 5-CYCLES
- Tan, Xiang (School of Mathematics and Quantitative Economics Shandong University of Finance and Economics)
- Received : 2015.01.05
- Published : 2016.01.31
A k-total-coloring of a graph G is a coloring of
planar graph;total coloring;adjacent 5-cycle
Supported by : National Natural Science Foundation of China, Natural Science Foundation of Shandong Province
- M. Behzad, Graphs and their chromatic numbers, Ph.D. thesis, Michigan State University, 1965.
- J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, New York, 1976.
- O. V. Borodin, A. V. Kostochka, and D. R. Woodall, Total colorings of planar graphs with large maximum degree, J. Graph Theory 26 (1997), no. 1, 53-59. https://doi.org/10.1002/(SICI)1097-0118(199709)26:1<53::AID-JGT6>3.0.CO;2-G
- O. V. Borodin, A. V. Kostochka, and D. R. Woodall, Total colorings of planar graphs with large girth, Europ. J. Combinatorics 19 (1998), 19-24. https://doi.org/10.1006/eujc.1997.0152
- J. S. Cai, Total coloring of a planar graph without 7-cycles with chords, Acta Math. Appl. Sin. 37 (2014), no. 2, 286-296.
- G. J. Chang, J. F. Hou, and N. Roussel, Local condition for planar graphs of maximum degree 7 to be 8-totally-colorable, Discrete Appl. Math. 159 (2011), no. 8, 760-768. https://doi.org/10.1016/j.dam.2011.01.001
- J. Chang, H. J. Wang, and J. L. Wu, Total colorings of planar graphs with maximum degree 8 and without 5-cycles with two chords, Theoret. Comput. Sci. 476 (2013), 16-23. https://doi.org/10.1016/j.tcs.2013.01.015
- D. Z. Du, L. Shen, and Y. Q. Wang, Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally colorable, Discrete Appl. Math. 157 (2009), no. 13, 2778-2784. https://doi.org/10.1016/j.dam.2009.02.011
- J. F. Hou, B. Liu, G. Z. Liu, and J. L. Wu, Total colorings of planar graphs without 6-cycles, Discrete Appl. Math. 159 (2011), no. 2-3, 157-163. https://doi.org/10.1016/j.dam.2010.08.025
- A. V. Kostochka, The total chromatic number of any multigraph with maximum degree five is at most seven, Discrete Math. 162 (1996), no. 1-3, 199-214. https://doi.org/10.1016/0012-365X(95)00286-6
- L. Kowalik, J. S. Sereni, and R. Skrekovski, Total-Coloring of plane graphs with maximum degree nine, SIAM J. Discrete Math. 22 (2008), no. 4, 1462-1479. https://doi.org/10.1137/070688389
- D. P. Sanders and Y. Zhao, On total 9-coloring planar graphs of maximum degree seven, J. Graph Theory 31 (1999), no. 1, 67-73. https://doi.org/10.1002/(SICI)1097-0118(199905)31:1<67::AID-JGT6>3.0.CO;2-C
- L. Shen and Y. Q. Wang, On the 7-total colorability of planar graphs with maximum degree 6 and without 4-cycles, Graphs Combin. 25 (2009), no. 3, 401-407. https://doi.org/10.1007/s00373-009-0843-y
- J. J. Tian, J. L. Wu, and H. J. Wang, Total colorings of planar graphs without adjacent chordal 5-cycles, Util. Math. 91 (2013), 13-23.
- V. G. Vizing, Some unresolved problems in graph theory, Uspekhi Mat. Nauk 23 (1968), 117-134.
- B. Wang and J. L. Wu, Total colorings of planar graphs without intersecting 5-cycles, Discrete Appl. Math. 160 (2012), no. 12, 1815-1821. https://doi.org/10.1016/j.dam.2012.03.027
- B. Wang, J. L. Wu, and H. J. Wang, Total colorings of planar graphs with maximum degree seven and without intersecting 3-cycles, Discrete Math. 311 (2011), no. 18-19, 2025-2030. https://doi.org/10.1016/j.disc.2011.05.038
- H. J. Wang, B. Liu, and J. L. Wu, Total colorings of planar graphs without adjacent 4-cycles, Discrete Math. 312 (2012), no. 11, 1923-1926. https://doi.org/10.1016/j.disc.2012.02.026
- H. J. Wang, L. D. Wu, and J. L. Wu, Total coloring of planar graphs with maximum degree 8, Theoret. Comput. Sci. 522 (2014), 54-61. https://doi.org/10.1016/j.tcs.2013.12.006
- H. J. Wang, L. D. Wu, W. L. Wu, P. M. Pardalos, and J. L. Wu, Minimum total coloring of planar graph, J. Global Optim. 60 (2014), no. 4, 777-791. https://doi.org/10.1007/s10898-013-0138-y
- P. Wang and J. Wu, A note on total colorings of planar graphs without 4-cycles, Discuss. Math. Graph Theory 24 (2004), no. 1, 125-135. https://doi.org/10.7151/dmgt.1219
- W. Wang, Total chromatic number of planar graphs with maximum degree ten, J. Graph Theory 54 (2007), no. 2, 91-102. https://doi.org/10.1002/jgt.20195
- Y. Q. Wang, Q. Sun, X. Tao, and L. Shen, Plane graphs with maximum degree 7 and without 5-cycles with chords are 8-totally-colorable, Sci. China Math. 41 (2011), no. 1, 95-104.
- H. P. Yap, Total colourings of graphs, Lecture Notes in Mathematics, Springer, 1623, 1996.