Abstract
A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K n with rb(n, H) number of colors contains a rainbow copy of H. Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdős et al. [4] and studied in numerous papers.
We show that \(rb(n, C_k^+) = rb(n, C_k)\) for n ≥ k + 1, where C + k denotes a cycle C k with a pendant edge.
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References
Alon, N.: On the conjecture of Erdős, Simonovits and Sós concerning anti-Ramsey theorems. J. Graph Theory 7, 91–94 (1983)
Diestel, R.: Graph theory. Springer-Verlag, New York, 1997
Erdős, P., Simonovits, M.: A limit theorem in graph theory, Studia Sci. Math. Hungar. 1, 51–57 (1966)
Erdős, P., Simonovits, A., Sós, V.: Anti–Ramsey theorems, Infinite and finite sets (A. Hajnal, R. Rado, and V.Sós, eds.), Colloq. Math. Soc. J. Bolyai, North-Holland, 1973, pp. 633–643
Gorgol, I.: On rainbow numbers for some graphs, preprint
Gorgol, I., Łazuka, E.: Rainbow numbers for small stars with one edge added, submitted, 2007
Jiang, T.: Anti-Ramsey numbers for subdivided graphs. J. Combin. Theory, Ser. B 85, 361–366 (2002)
Jiang, T.: Edge-colorings with no large polychromatic stars. Graphs and Combinatorics 18, 303–308 (2002)
Jiang, T., West, D.B.: On the Erdős-Simonovits-Sós conjecture about the anti-Ramsey number of a cycle. Combin. Probab. Comput. 12, 585–598 (2003)
Jiang, T., West, D.B.: Edge-colorings of complete graphs that avoid polychromatic trees. Discrete Math. 274, 137–145 (2004)
Montellano-Ballesteros, J.J., Neuman-Lara, V.: An anti–Ramsey theorem on cycles. Graphs and Combinatorics 21(3), 343–354 (2005)
Ore, O.: Arc coverings of graphs. Ann. Math. Pure Appl. 55, 315–321 (1961)
Schiermeyer, I.: Rainbow 5- and 6-cycles: a proof of the conjecture of Erdős, Simonovits and Sós, preprint, TU Bergakademie Freiberg, 2001
Schiermeyer, I.: Rainbow numbers for matchings and complete graphs. Discrete Math. 286, 157–162 (2004)
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Gorgol, I. Rainbow Numbers for Cycles with Pendant Edges. Graphs and Combinatorics 24, 327–331 (2008). https://doi.org/10.1007/s00373-008-0786-8
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DOI: https://doi.org/10.1007/s00373-008-0786-8