Abstract
An edge colored graph is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number, rc-number for short, of a graph \({\varGamma }\), is the smallest number of colors that are needed in order to make \({\varGamma }\) rainbow connected. In this paper, we give a method to bound the rc-numbers of graphs with certain structural properties. Using this method, we investigate the rc-numbers of Cayley graphs, especially, those defined on abelian groups and on dihedral groups.
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References
Akers SB, Krishnamurthy B (1989) A group-theoretic model for symmetric interconnection networks. IEEE Trans Comput 38:555–566
Basavaraju M, Chandran LS, Rajendraprasad D, Ramaswamy A (2014) Rainbow connection number and radius. Graphs Combin 30(2):275–285
Bondy JA, Murty USR (2008) Graph theory. Springer, Berlin
Cai QQ, Ma YB, Song JL. Rainbow connection numbers of ladders and Möbius ladders. Ars Combin (to appear)
Caro Y, Lev A, Roditty Y, Tuza Z, Yuster R (2008) On rainbow connection. Electron J Combin 15:R57
Chakraborty S, Fischer E, Matsliah A, Yuster R (2011) Hardness and algorithms for rainbow connection. J Comb Optim 21:330–347
Chandran LS, Das A, Rajendraprasad D, Varma NM (2012) Rainbow connection number and connected dominating sets. J Graph Theory 71:206–218
Chartrand G, Johns GL, McKeon KA, Zhang P (2008) Rainbow connection in graphs. Math Bohem 133:85–98
Godsil C, Royle G (2001) Algebraic graph theory. Springer, New York
Krivelevich M, Yuster R (2009) The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J Graph Theory 63:185–191
Li HZ, Li XL, Liu SJ (2011) The (strong) rainbow connection numbers of Cayley graphs on Abelian groups. Comput Math Appl 62:4082–4088
Li XL, Liu SJ, Chandran LS, Mathew R, Rajendraprasad D (2012) Rainbow connection number and connectivity. Electron J Combin 19:R20
Li XL, Sun YF (2012) Rainbow connections of graphs. Springer, New York
Liang YJ (2012) Rainbow connection numbers of Cartesian product of graphs. 2012 Workshop on Graph Theory and Combinatorics and 2012 Symposium for Young Combiantorialists, August, pp. 10–12
Schiermeyer I (2009) Rainbow connection in graphs with minimum degree three, IWOCA 2009. LNCS, vol 5874. Springer, Berlin, pp 432–437
Acknowledgments
The authors are very grateful to the referees for helpful comments and suggestions. This work partially supported by the NSFC (Nos. 11271267, 11371204, 11526082, 11526081 and 11501181), and the Scientific Research Foundation for Ph.D. of Henan Normal University (No. qd14143).
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Ma, Y., Lu, Z. Rainbow connection numbers of Cayley graphs. J Comb Optim 34, 182–193 (2017). https://doi.org/10.1007/s10878-016-0052-6
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DOI: https://doi.org/10.1007/s10878-016-0052-6