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Gallai–Ramsey Numbers for a Class of Graphs with Five Vertices

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A Correction to this article was published on 03 October 2020

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Abstract

Given two graphs G and H, the k-colored Gallai–Ramsey number \(gr_k(G : H)\) is defined to be the minimum integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we consider \(gr_k(K_3 : H)\), where H is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai–Ramsey numbers for eight of them have been studied step by step in several papers. We determine all the Gallai–Ramsey numbers for the remaining five graphs, and we also obtain some related results for a class of unicyclic graphs. As applications, we find the mixed Ramsey spectra \(S(n; H, K_3)\) for these graphs by using the Gallai–Ramsey numbers.

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  • 03 October 2020

    The following items are the errors that should be corrected.

References

  1. Axenovich, M., Choi, J.: On colorings avoiding a rainbow cycle and a fixed monochromatic subgraph. Electron. J. Combin. 17(Research Paper 31), 15 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Axenovich, M., Choi, J.: A note on the monotonicity of mixed Ramsey numbers. Discret. Math. 311, 2020–2023 (2011)

    Article  MathSciNet  Google Scholar 

  3. Burr, S.A., Roberts, J.A.: On Ramsey numbers for stars. Utilitas Math. 4, 217–220 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Chen, M., Li, Y.S., Pei, C.P.: Gallai-Ramsey numbers of odd cycles and complete bipartite graphs. Graphs Combin. 34, 1185–1196 (2018)

    Article  MathSciNet  Google Scholar 

  5. Chung, F.R.K., Graham, R.: Edge-colored complete graphs with precisely colored subgraphs. Combinatorica 3, 315–324 (1983)

    Article  MathSciNet  Google Scholar 

  6. Faudree, R.J., Gould, R., Jacobson, M., Magnant, C.: Ramsey numbers in rainbow triangle free colorings. Aust. J. Combin. 46, 269–284 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Fujita, S., Magnant, C.: Gallai–Ramsey numbers for cycles. Discret. Math. 311, 1247–1254 (2011)

    Article  MathSciNet  Google Scholar 

  8. Fujita, S., Magnant, C.: Extensions of Gallai–Ramsey results. J. Graph Theory 70, 404–426 (2012)

    Article  MathSciNet  Google Scholar 

  9. Fujita, S., Magnant, C., Ozeki, K.: Rainbow generalizations of Ramsey theory: a survey. Graphs Combin. 26, 1–30 (2010)

    Article  MathSciNet  Google Scholar 

  10. Fujita, S., Magnant, C., Ozeki, K.: Rainbow generalizations of Ramsey theory: a dynamic survey. Theory Appl. Graphs 0, Article 1 (2014)

  11. Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hungar. 18, 25–66 (1967)

    Article  MathSciNet  Google Scholar 

  12. Gyárfás, A., Sárközy, G.N., Sebő, A., Selkow, S.: Ramsey-type results for Gallai colorings. J. Graph Theory 64, 233–243 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Gyárfás, A., Simonyi, G.: Edge colorings of complete graphs without tricolored triangles. J. Graph Theory 46, 211–216 (2004)

    Article  MathSciNet  Google Scholar 

  14. Hall, M., Magnant, C., Ozeki, K., Tsugaki, M.: Improved upper bounds for Gallai–Ramsey numbers of paths and cycles. J. Graph Theory 75, 59–74 (2014)

    Article  MathSciNet  Google Scholar 

  15. Hendry, G.R.T.: Ramsey numbers for graphs with five vertices. J. Graph Theory 13, 245–248 (1989)

    Article  MathSciNet  Google Scholar 

  16. Li, X.H., Wang, L.G., Liu, X.X.: Complete graphs and complete bipartite graphs without rainbow path. Discret. Math. 342, 2116–2126 (2019)

    Article  MathSciNet  Google Scholar 

  17. Maffray, F., Preissmann, M.: A translation of Tibor Gallai’s paper: transitiv orientierbare Graphen. In: Ramirez-Alfonsin, J.L., Reed, B.A. (eds.) Perfect Graphs, pp. 25–66. Wiley, New York (2001)

    MATH  Google Scholar 

  18. Mao, Y.P., Wang, Z., Magnant, C., Schiermeyer, I.: Gallai–Ramsey numbers for fans. arXiv:1902.10706

  19. Radziszowski, S.P.: Small Ramsey numbers. Electron J. Combin. 1, Dynamic Survey 1, (electronic) (2017)

  20. Song, Z.-X., Zhang, J.M.: A conjecture on Gallai–Ramsey numbers of even cycles and paths. arXiv:1803.07963

  21. Wang, Z., Mao, Y.P., Magnant, C., Zou, J.Y.: Ramsey and Gallai–Ramsey numbers for two classes of unicyclic graphs. arXiv:1809.10298

  22. Wu, H.B., Magnant, C., Nowbandegani, P.S., Xia, S.M.: All partitions have small parts-Gallai-Ramsey numbers of bipartite graphs. Discret. Appl. Math. 254, 196–203 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments, suggestions and corrections which improved the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Xihe Li & Ligong Wang

  2. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Xihe Li & Ligong Wang

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  1. Xihe Li
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  2. Ligong Wang
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Correspondence to Ligong Wang.

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Supported by the National Natural Science Foundation of China (No. 11871398), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JM1032), the Fundamental Research Funds for the Central Universities (No. 3102019ghjd003), and China Scholarship Council (No. 201906290174).

The original version of this article was revised: In the Original publication of the article, few errors have occurred. The corrections are given below:

1. Page 3, Theorem 2:

\( gr_k(K_3 : F_9)=gr_k(K_3 : F_{10})= \left\{ \begin{array}{l} 8\cdot 5^{(k-2)/2}+1,\\ 4\cdot 5^{(k-1)/2}+1,. \end{array} \right. \)

should be

\( gr_k(K_3 : F_9)=gr_k(K_3 : F_{10})= \left\{ \begin{array}{ll} 8\cdot 5^{(k-2)/2}+1,& \text{ if } k \text{ is } \text{ even},\\ 4\cdot 5^{(k-1)/2}+1,& \text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

2. Page 3, Theorem 4 (3):

\(k(n-1)+2 \ge gr_k(K_3 : F_{2,n})\ge \left\{ \begin{array} {l} \frac{5n}{2}+k-6, \\ \frac{5n-1}{2}+k-4, \end{array} \right. \)

should be

\(k(n-1)+2 \ge gr_k(K_3 : F_{2,n})\ge \left\{ \begin{array} {ll} \frac{5n}{2}+k-6,& \text{ if } n \text{ is } \text{ even},\\ \frac{5n-1}{2}+k-4,& \text{ if } n \text{ is } \text{ odd}. \end{array} \right. \)

3. Page 4, Theorem 5:

\( gr_k(K_3 : F_{12})=gr_k(K_3 : F_{13})= \left\{ \begin{array}{l} 9\cdot 5^{(k-2)/2}+1, \\ 4\cdot 5^{(k-1)/2}+1, \end{array} \right.\)

should be

\( gr_k(K_3 : F_{12})=gr_k(K_3 : F_{13})= \left\{ \begin{array}{ll} 9\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

4. Page 5, Lemma 2:

\(gr_k(K_3 : K_3)= \left\{ \begin{array}{l}5^{k/2}+1,\\2\cdot 5^{(k-1)/2}+1, \end{array} \right.\)

should be

\(gr_k(K_3 : K_3)= \left\{ \begin{array}{ll} 5^{k/2}+1,&\text{ if } k \text{ is } \text{ even},\\2\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right.\)

5. Page 5, Lemma 3:

\(gr_k(K_3 : F_9)> \left\{ \begin{array}{l}8\cdot 5^{(k-2)/2},\\4\cdot 5^{(k-1)/2}, \end{array} \right.\)

should be

\(gr_k(K_3 : F_9)> \left\{ \begin{array}{ll}8\cdot 5^{(k-2)/2},&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2},&\text{ if } k \text{ is } \text{ odd}. \end{array} \right.\)

6. Page 5, Lemma 4:

\(gr_k(K_3 : F_{10})\le \left\{ \begin{array}{l}8\cdot 5^{(k-2)/2}+1,\\4\cdot 5^{(k-1)/2}+1, \end{array} \right.\)

should be

\(gr_k(K_3 : F_{10})\le \left\{ \begin{array}{ll}8\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right.\)

7. Page 5, Proof of lemma 4:

\(n = \left\{ \begin{array}{l}8\cdot 5^{(k-2)/2}+1,\\4\cdot 5^{(k-1)/2}+1, \end{array} \right.\)

should be

\(n = \left\{ \begin{array}{ll}8\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right.\)

8. Page 6:

\(gr_{k-1}(K_3 : K_3)= \left\{ \begin{array}{l}2\cdot 5^{(k-2)/2}+1,\\5^{(k-1)/2}+1, \end{array} \right.\)

should be

\(gr_{k-1}(K_3 : K_3)= \left\{ \begin{array}{ll}2\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

9. Page 9:

\( |V(G_k)|= \left\{ \begin{array}{l} \frac{5n}{2}+k-7,\\ \frac{5n-1}{2}+k-5, \end{array} \right. \)

should be

\( |V(G_k)|= \left\{ \begin{array}{ll}\frac{5n}{2}+k-7,&\text{ if } n \text{ is } \text{ even},\\ \frac{5n-1}{2}+k-5,&\text{ if } n \text{ is } \text{ odd}. \end{array} \right. \)

10. Page 9:

\( n_{k}= \left\{ \begin{array}{l} r_2(F_{2,n})+k-2,\\ k+9,\\k(n-1)+2, \end{array} \right. \)

should be

\( n_{k}= \left\{ \begin{array}{lll}r_2(F_{2,n})+k-2,&\text{ if } n \in \{3,4\} \text{ and } k\ge 1,\\k+9,&\text{ if } n=5 \text{ and } k\ge 2,\\k(n-1)+2,&\text{ if } n \ge 6 \text{ and } k\ge 2. \end{array} \right. \)

11. Page 11, Lemma 6:

\( gr_k(K_3 : H)> \left\{ \begin{array}{l}9\cdot 5^{(k-2)/2},\\4\cdot 5^{(k-1)/2}, \end{array} \right. \)

should be

\( gr_k(K_3 : H)> \left\{ \begin{array}{ll}9\cdot 5^{(k-2)/2},&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2},&\text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

12. Page 11–12, Lemma 7:

\( gr_k(K_3 : H)\le \left\{ \begin{array}{l}9\cdot 5^{(k-2)/2}+1,\\4\cdot 5^{(k-1)/2}+1, \end{array} \right. \)

should be

\(gr_k(K_3 : H)\le \left\{ \begin{array}{ll}9\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

13. Page 12, Proof of Lemma 7:

\( n= \left\{ \begin{array}{l}9\cdot 5^{(k-2)/2}+1,\\4\cdot 5^{(k-1)/2}+1, \end{array} \right. \)

should be

\( n= \left\{ \begin{array}{ll}9\cdot 5^{(k-2)/2}+1,&\text{ if } k \text{ is } \text{ even},\\4\cdot 5^{(k-1)/2}+1,&\text{ if } k \text{ is } \text{ odd}. \end{array} \right. \)

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Li, X., Wang, L. Gallai–Ramsey Numbers for a Class of Graphs with Five Vertices. Graphs and Combinatorics 36, 1603–1618 (2020). https://doi.org/10.1007/s00373-020-02194-5

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