Abstract
Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is also called total coloring. We consider a planar graph G with maximum degree Δ(G) ≥ 8, and proved that if G contains no adjacent i, j-cycles with two chords for some i, j ∈ {5, 6, 7}, then G is total-(Δ + 1)-colorable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arroyo, S.: Determing the total colouring number is NP-hard. Discrete Math., 78, 315–319 (1989)
Behzad, M.: Graphs and Their Chromatic Numbers, Ph.D. Thesis, Michigan State University, 1965
Bojarshinov, V. A.: Edge and total coloring of interal graphs. Discrete Appl. Math., 114, 23–28 (2001)
Bondy, J. A. U., Murty, S. R.: Graph Theory with Applications, MacMillan, London, 1976
Chang, J., Wang, H. J., Wu, J. L., et al.: Total colorings of planar graphs with maximum degree 8 and without 5-cycles with two chords. Theoret. Comput. Sci., 476, 16–23 (2013)
Du, D. Z., Shen, L., Wang, Y. Q.: Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable. Discrete Appl. Math., 157, 2778–2784 (2009)
Kostochka, A. V.: The total chromatic number of any multigraph with maximum degree five is at most seven. Discrete Math., 162, 199–214 (1996)
Kowalik, L., Sereni, J. S., Skrekovski, R.: Total-Coloring of plane graphs with maximum degree nine. SIAM J. Discrete Math., 22, 1462–1479 (2008)
McDiarmid, C. J. H., Arroyo, S.: Total colorings regular bipartite graphs is NP-hard. Discrete Math., 124, 155–162 (1994)
Sanders, D. P., Zhao, Y.: On total 9-coloring planar graphs of maximum degree seven. J. Graph Theory, 31, 67–73 (1999)
Tian, J. J., Wu, J. L., Wang, H. J.: Total coloring of planar graphs without adjacent chordal 5-cycles. Util. Math., 91, 13–23 (2013)
Vizing, V. G.: Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk, 23, 117–134 (1968)
Wang, H. J., Liu, B., Wu, J. L.: Total coloring of planar graphs without chordal short cycles. Graphs Combin., 31, 1755–1764 (2015)
Wang, H. J., Wu, L. D., Wu, W. L., et al.: Minimum total coloring of planar graph. J. Global Optim., 60, 777–791 (2014)
Wu, J. L.: Total coloring of series-parallel graphs. Ars Combin., 73, 215–217 (2004)
Xu, R. Y., Wu, J. L., Wang, H. J.: Total colouring of planar graphs without some chordal 6-cycles. Bull. Malays. Math. Sci. Soc., 38, 561–569 (2015)
Yap, H. P.: Total colourings of graphs, Lecture Notes in Mathematics, vol. 1623, Springer, Berlin, 1996
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grants Nos. 11401386, 11402075, 11501316, 71171120 and 71571180), China Postdoctoral Science Foundation (Grants Nos. 2015M570568, 2015M570572), the Qingdao Postdoctoral Application Research Project (Grants Nos. 2015138, 2015170), the Shandong Provincial Natural Science Foundation of China (Grants Nos. ZR2013AM001, ZR2014AQ001, ZR2015GZ007, ZR2015FM023), the Scientific Research Program of the Higher Education Institution of Xinjiang (Grant No. XJEDU20141046)
Rights and permissions
About this article
Cite this article
Wang, H.J., Luo, Z.Y., Liu, B. et al. A note on the minimum total coloring of planar graphs. Acta. Math. Sin.-English Ser. 32, 967–974 (2016). https://doi.org/10.1007/s10114-016-5427-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-5427-1