TOTAL COLORINGS OF PLANAR GRAPHS WITH MAXIMUM DEGREE AT LEAST 7 AND WITHOUT ADJACENT 5-CYCLES

Title & Authors
TOTAL COLORINGS OF PLANAR GRAPHS WITH MAXIMUM DEGREE AT LEAST 7 AND WITHOUT ADJACENT 5-CYCLES
Tan, Xiang;

Abstract
A k-total-coloring of a graph G is a coloring of $\small{V{\cup}E}$ using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number $\small{{\chi}^{{\prime}{\prime}}(G)}$ of G is the smallest integer k such that G has a k-total-coloring. Let G be a planar graph with maximum degree $\small{{\Delta}}$. In this paper, it`s proved that if $\small{{\Delta}{\geq}7}$ and G does not contain adjacent 5-cycles, then the total chromatic number $\small{{\chi}^{{\prime}{\prime}}(G)}$ is $\small{{\Delta}+1}$.
Keywords
Language
English
Cited by
References
1.
M. Behzad, Graphs and their chromatic numbers, Ph.D. thesis, Michigan State University, 1965.

2.
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, New York, 1976.

3.
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.

4.
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.

5.
J. S. Cai, Total coloring of a planar graph without 7-cycles with chords, Acta Math. Appl. Sin. 37 (2014), no. 2, 286-296.

6.
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.

7.
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.

8.
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.

9.
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.

10.
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.

11.
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.

12.
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.

13.
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.

14.
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.

15.
V. G. Vizing, Some unresolved problems in graph theory, Uspekhi Mat. Nauk 23 (1968), 117-134.

16.
B. Wang and J. L. Wu, Total colorings of planar graphs without intersecting 5-cycles, Discrete Appl. Math. 160 (2012), no. 12, 1815-1821.

17.
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.

18.
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.

19.
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.

20.
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.

21.
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.

22.
W. Wang, Total chromatic number of planar graphs with maximum degree ten, J. Graph Theory 54 (2007), no. 2, 91-102.

23.
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.

24.
H. P. Yap, Total colourings of graphs, Lecture Notes in Mathematics, Springer, 1623, 1996.