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

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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.

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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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