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

1 Correction to: Graphs and Combinatorics https://doi.org/10.1007/s00373-020-02194-5

In the original publication of the article, few errors have occurred. The corrections are given below:

• 1. Page 3, Theorem 2:

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

  should be

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

• 2. Page 3, Theorem 4 (3):

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

  should be

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

• 3. Page 4, Theorem 5:

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

  should be

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

• 4. Page 5, Lemma 2:

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

  should be

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

• 5. Page 5, Lemma 3:

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

  should be

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

• 6. Page 5, Lemma 4:

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

  should be

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

• 7. Page 5, Proof of lemma 4:

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

  should be

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

• 8. Page 6:

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

  should be

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

• 9. Page 9:

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

  should be

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

• 10. Page 9:

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

  should be

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

• 11. Page 11, Lemma 6:

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

  should be

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

• 12. Page 11–12, Lemma 7:

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

  should be

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

• 13. Page 12, Proof of Lemma 7:

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

  should be

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

The original article has been corrected.