## The description of the correction

Authors have identified errors in the paper originally submitted and finally approved (after the acceptance) by the Authors.

*On page 74, case 2 second line and third line highlighted part to be changed.*

**Case 2: **$\mathit{n}\ge 3$

By Proposition 2.2, $\chi \left[T\left({P}_{n}\right)\right]\ge 3$ as $T\left({P}_{n}\right)$ includes an odd cycle. Assign the proper coloring to the vertices as $f\left({v}_{i}\right)=\mathrm{1,3},\mathrm{2,1},\mathrm{3,2},\dots .,n$, $f\left({u}_{i}\right)=\mathrm{2,1},\mathrm{3,2},\mathrm{1,3},\dots .,n-1$. Thus, a minimum of three colors are required for proper coloring. Therefore $\chi \left[T\left({P}_{n}\right)\right]=3$.

*On page 74, an error in the symbol of statement of Theorem 3.3.*

**Theorem 3.3**

*On page 75, third paragraph second line of case when *$n$* = 6, 4 is to be written as 3.*

In this case the set $\{{v}_{1},{v}_{6},{u}_{4}\}$ or $\{{v}_{2},{v}_{5},{u}_{3}\}$ are only $\gamma $-sets of graph $T\left({P}_{6}\right)$. According to Lemma – 3.1, $\gamma \left[T\left({P}_{6}\right)\right]=3$ and by Lemma – 3.2, $\chi \left[T\right({P}_{6}\left)\right]=3$. Allocating several colors to the vertices of the $\gamma $-set that is equal to $\gamma \left[T\left({P}_{6}\right)\right]$ in order to determine its optimal coloring. Now we use $\chi \left[T\right({P}_{6}\left)\right]-1$ number of colors to color the remaining vertices.

*On page 75, Case 1 last line in place of 6, it should be 5. *

The coloring pattern can be defined as $f\left({v}_{1}\right)=f\left({v}_{4}\right)=f\left({u}_{2}\right)=f\left({u}_{5}\right)=3,f\left({v}_{3}\right)=1,f\left({v}_{6}\right)=f\left({u}_{1}\right)=f\left({u}_{4}\right)=4,f\left({v}_{2}\right)=1,f\left({u}_{4}\right)=2,f\left({v}_{5}\right)=2,f\left({u}_{3}\right)=5.$ Here every vertex dominates the vertices of at least one color class. As a result, the proper coloring creates a dominator coloring for the relevant graph. Therefore, ${\chi}_{d}\left[T\left({P}_{6}\right)\right]=5=\chi \left[T\left({P}_{6}\right)\right]+\gamma \left[T\left({P}_{6}\right)\right]-1$.

*On page 75, Case 2 title is mentioned incorrect. *

**Case 2: **$\mathit{n}\ge 5$**, **$\mathit{n}\ne 6$

*On page 75, In the line just above the Fig. 1, in place of six colors there must be five colors.*

A dominator coloring of $T\left({P}_{6}\right)$ using five colors is shown in Fig. 1.