Erratum. Dominator coloring of total graph of path and cycles

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.