Abstract
The clique-chromatic number of a graph is the least number of colors on the vertices of the graph so that no maximal clique of size at least two is monochromatic. In 2003, Gravier, Hoang, and Maffray have shown that, for any graph \(F\), the class of \(F\)-free graphs has a bounded clique-chromatic number if and only if \(F\) is a vertex-disjoint union of paths, and they give an upper bound for all such cases. In this paper, their bounds for \(F=P_2+kP_1\) and \(F=P_3+kP_1\) with \(k \ge 3\) are significantly reduced to \(k+1\) and \(k+2\) respectively, and sharp bounds are given for some subclasses.
Tanawat Wichianpaisarn—Partially supported by His Royal Highness Crown Prince Maha Vajiralongkorn Fund.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bacsó, G., Gravier, S., Gyárfás, A., Preissmann, M., Sebő, A.: Coloring the maximal cliques of graphs. SIAM J. Discrete Math. 17, 361–376 (2004)
Campos, C.N., Dantasa, S., de Mello, C.P.: Colouring clique-hypergraphs of circulant graphs. Electron. Notes Discrete Math. 30, 189–194 (2008)
Defossez, D.: Clique-coloring some classes of odd-hole-free. J. Graph Theory 53, 233–249 (2006)
Duffus, D., Sands, B., Sauer, N., Woodrow, R.E.: Two-coloring all two-element maximal antichains. J. Comb. Theory Ser. A 57, 109–116 (1991)
Duffus, D., Kierstead, H.A., Trotter, W.T.: Fibres and ordered set coloring. J. Comb. Theory Ser. A 58, 158–164 (1991)
Gravier, S., Hoáng, C.T., Maffray, F.: Coloring the hypergraph of maximal cliques of a graph with no long path. Discrete Math. 272, 285–290 (2003)
Gravier, S., S̆krekovski, R.: Coloring the clique hypergraph of graphs without forbidden structure, Les cahiers du laboratoire Leibniz 83, (2003). http://www-leibniz.imag.fr/LesCahiers/
Mycielski, J.: Sur le coloriage des graphes. J. Colloq. Math. 3, 161–162 (1955)
West, D.B.: Introduction to Graph Theory. Prentice Hall, New Jersey (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Wichianpaisarn, T., Uiyyasathian, C. (2014). More Results on Clique-chromatic Numbers of Graphs with No Long Path. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-13287-7_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-13286-0
Online ISBN: 978-3-319-13287-7
eBook Packages: Computer ScienceComputer Science (R0)