Abstract
Let \(G\) be a planar graph with \(\varDelta \ge 8\) and \(v\) be a vertex of \(G\). It is proved that \(\chi ''(G)=\varDelta +1\) if \(v\) is not incident with chordal \(5\)- or \(6\)-cycles by Shen et al. (Appl Math Lett 22:1369–1373, 2009), or \(v\) is not incident with \(2\)-chordal \(5\)-cycle by Chang et al. (Theor Comput Sci 476:16–23, 2013). In this paper we generalize these results and prove that if \(v\) is not incident with chordal \(6\)-cycle, or chordal \(7\)-cycle, or \(2\)-chordal \(5\)-cycle, then \(\chi ''(G)=\varDelta +1\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borodin, O.V.: On the total coloring of planar graphs. J. Reine Angew. Math. 394, 180–185 (1989)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. MacMillan, London (1976)
Chang, G.J., Hou, J.F., Roussel, N.: Local condition for planar graphs of maximum degree 7 to be 8-totally colorable. Discrete Appl. Math. 159, 760–768 (2011)
Chang, J., Wang, H.J., Wu, J.L., A, Y.G.: Total colorings of planar graphs with maximum degree 8 and without 5-cycles with two chords. Theor. 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., S̆krekovski, R.: Total-coloring of plane graphs with maximum degree nine. SIAM J. Discrete Math. 22, 1462–1479 (2008)
Hou, J.F., Liu, B., Liu, G.Z., Wu, J.L.: Total colorings of planar graphs without 6-cycles. Discrete Appl. Math. 159, 157–163 (2011)
Liu, B., Hou, J.F., Wu, J.L., Liu, G.Z.: Total colorings and list total colorings of planar graphs without intersecting 4-cycles. Discrete Math. 309, 6035–6043 (2009)
Roussel, N., Zhu, X.: Total coloring of planar graphs of maximum degree eight. Inf. Process. Lett. 110, 321–324 (2010)
Sanders, D.P., Zhao, Y.: On total 9-coloring planar graphs of maximum degree seven. J. Graph. Theory 31, 67–73 (1999)
Shen, L., Wang, Y.Q.: Total colorings of planar graphs with maximum degree at least 8. Sci. China Ser. A 52(8), 1733–1742 (2009)
Shen, L., Wang, Y.Q., et al.: On the 9-total-colorability of planar graphs with maximum degree 8 and without intersecting triangles. Appl. Math. Lett. 22, 1369–1373 (2009)
Sun, X.Y., Wu, J.L., Wu, Y.W., Hou, J.F.: Total colorings of planar graphs without adjacent triangles. Discrete Math. 309, 202–206 (2009)
Wang, H.J., Wu, J.L.: A note on the total coloring of planar graphs without adjacent 4-cycles. Discrete Math. 312, 1923–1926 (2012)
Wang, H.J., Liu, B., Wu, J.L., Wang, B.: Total coloring of graphs embedded in surfaces of nonnegative Euler characteristic. Sci. China. Math. 57 (1), 211–220 (2014)
Wang, W.F.: Total chromatic number of planar graphs with maximum degree ten. J. Graph. Theory 54(2), 91–102 (2007)
Wu, J.L., Wang, P.: List-edge and list-total colorings of graphs embedded on hyperbolic surfaces. Discrete Math. 308(24), 6210–6215 (2008)
Yap, H.P.: Total Colourings of Graphs. Lecture Notes in Mathematics, vol. 1623. Springer, Berlin (1996)
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants 11201440, 11271006, 11271341, 11301410, the Scientific Research Foundation for the Excellent Young and Middle-Aged Scientists of Shandong Province of China under Grant BS2013DX002, and the Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2013JQ1002.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Liu, B. & Wu, J. Total Coloring of Planar Graphs Without Chordal Short Cycles. Graphs and Combinatorics 31, 1755–1764 (2015). https://doi.org/10.1007/s00373-014-1449-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1449-6