Abstract
A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacsó et al. (SIAM J Discrete Math 17:361–376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839–938, 2008) and the limitation of the degree 7 is necessary since the line graph of \(K_{6}\) is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number \(R(3,3)=6\). In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable (\(m\ge 3\)), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable.
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This research was partially supported by the National Nature Science Foundation of China (Nos. 11426144 and 11571222) and the Nature Science Foundation of Shandong Province, China (No. ZR2014AQ008).
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Liang, Z., Shan, E. & Kang, L. Clique-Coloring Claw-Free Graphs. Graphs and Combinatorics 32, 1473–1488 (2016). https://doi.org/10.1007/s00373-015-1657-8
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DOI: https://doi.org/10.1007/s00373-015-1657-8