Published: 21 August 2023

Erratum. Dominator coloring of total graph of path and cycles

Minal Shukla1
Foram Chandarana2
1, 2Marwadi University, Rajkot, Gujarat, 360003, India
Corresponding Author:
Minal Shukla
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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: n3

By Proposition 2.2, χTPn3 as TPn includes an odd cycle. Assign the proper coloring to the vertices as fvi=1,3,2,1,3,2,.,n, fui=2,1,3,2,1,3,.,n-1. Thus, a minimum of three colors are required for proper coloring. Therefore χTPn=3.

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

Theorem 3.3

χdTPn=χTPn+γTPn-1,n=2,3,4,6,χTPn+γTPn,n5, n6.

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

In this case the set {v1,v6,u4} or {v2,v5,u3} are only γ-sets of graph T(P6). According to Lemma – 3.1, γT(P6)=3 and by Lemma – 3.2, χ[T(P6)]=3. Allocating several colors to the vertices of the γ-set that is equal to γT(P6) in order to determine its optimal coloring. Now we use χ[T(P6)]-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 fv1=fv4=fu2=fu5=3, fv3=1, fv6=fu1=fu4=4,fv2=1,fu4=2,fv5=2,f(u3)=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, χdT(P6)=5=χTP6+γTP6-1.

On page 75, Case 2 title is mentioned incorrect.

Case 2: n5, n6

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 TP6 using five colors is shown in Fig. 1.

About this article

07 August 2023
07 August 2023
21 August 2023