Graphs and Combinatorics

, Volume 31, Issue 6, pp 1985–1991

# Rainbow Numbers for Graphs Containing Small Cycles

Original Paper

## Abstract

For a given graph $$H$$ and $$n \ge 1,$$ let $$f(n, H)$$ denote the maximum number $$m$$ for which it is possible to colour the edges of the complete graph $$K_n$$ with $$m$$ colours in such a way that each subgraph $$H$$ in $$K_n$$ has at least two edges of the same colour. Equivalently, any edge-colouring of $$K_n$$ with at least $$rb(n,H)=f(n,H) + 1$$ colours contains a rainbow copy of $$H.$$ The numbers $$f(n,H)$$ and $$rb(K_n,H)$$ are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph $$H$$ with respect to its cyclomatic number. Let $$H$$ be a graph of order $$p \ge 4$$ and cyclomatic number $$v(H) \ge 2.$$ Then $$rb(K_n, H)$$ cannot be bounded from above by a function which is linear in $$n.$$ If $$H$$ has cyclomatic number $$v(H) = 1,$$ then $$rb(K_n, H)$$ is linear in $$n.$$ Moreover, we will compute all rainbow numbers for the bull $$B,$$ which is the unique graph with $$5$$ vertices and degree sequence $$(1,1,2,3,3)$$.

### Keywords

Rainbow colouring Rainbow number Anti-ramsey number Turán number

## Notes

### Acknowledgments

We thank Jana Neupauerová for some stimulating discussions on the computation of the rainbow numbers for the bull and an anonymous referee for some valuable comments.

### References

1. 1.
Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Macmillan, London and Elsevier, New York (1976)Google Scholar
2. 2.
Erdős, P.: Graph theory and probability. Can. J. Math. 11, 34–38 (1959)
3. 3.
Erdős, P., Simonovits, M., Sós, V.T.: Anti-Ramsey theorems, infinite and finite sets, vol. II. In: Hajnal, A., Rado, R., Sós, V.T. (eds.) Colloq. Math. Soc. János Bolyai 10, pp. 633–643 (1975)Google Scholar
4. 4.
Fujita, S., Kaneko, A., Schiermeyer, I., Suzuki, K.: A rainbow k-matching in the complete graph with r colors. Electron. J. Comb. 16, R51 (2009)
5. 5.
Fujita, S., Magnant, C., Ozeki, K.: Rainbow generalizations of Ramsey theory—a dynamic survey. Theory Appl. Graphs (1) (Article 1) (2014)Google Scholar
6. 6.
Gorgol, I.: Rainbow numbers for cycles with pendant edges. Graphs Comb. 24, 327–331 (2008)
7. 7.
Montellano-Ballesteros, J.J.: An anti-Ramsey theorem on diamonds. Graphs Comb. 26, 283–291 (2010)
8. 8.
Montellano-Ballesteros, J.J., Neumann-Lara, V.: An anti-Ramsey theorem. Combinatorica 22(3), 445–449 (2002)
9. 9.
Montellano-Ballesteros, J.J., Neumann-Lara, V.: An anti-Ramsey theorem on cycles. Graphs Comb. 21(3), 343–354 (2005)
10. 10.
Schiermeyer, I.: Rainbow numbers for matchings and complete graphs. Discrete Math. 286, 157–162 (2004)
11. 11.
Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436–452 (1941)

## Authors and Affiliations

1. 1.Institut für Diskrete Mathematik und AlgebraTechnische Universität Bergakademie FreibergFreibergGermany
2. 2.Institute of MathematicsP.J. Šafárik University in KošiceKosiceSlovakia