Correction: Latchoumanane, V.; Varadhan, M. Antimagic Labeling for Product of Regular Graphs. Symmetry 2022, 14, 1235

Vinothkumar Latchoumanane; Murugan Varadhan

doi:10.3390/sym18030491

In Abstract and Keywords, Text correction:

Abstract: An antimagic labeling of a graph $G = (V, E)$ is a bijection from the set of edges of G to $\{1, 2, \dots, |E (G)|\}$ and such that any two vertices of G have distinct vertex sums where the vertex sum of a vertex v in $V (G)$ is nothing but the sum of all the incident edge labeling of G. In this paper, we discussed the antimagicness for rooted product of regular graph and for certain condition the corona product of regular graphs is antimagic. We proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the rooted product of graph G and H admits antimagic labeling for $k \geq c n$ for some $c \geq 1 .$ We also proved that, let G be a connected graph and $H$ be a sequence of regular graphs under certain conditions admits an antimagic labeling.

Keywords: graph labeling; antimagic labeling; product graphs; rooted product; corona product; regular graph

In Section 1. Introduction:

Missing Citation: In the original publication [1], the reference [2] was not cited. The citation has now been inserted as reference [15] in the paragraph 5 of the Introduction. The reference [3,4,5] was not cited. This citation is now been inserted as references [30–32] respectively, in the second paragraph after Theorem 1.

Text removed: In the original publication [1], the paragraph 4 is removed.

In Section 2. Main Results:

Text included: Under Observation 1, lemma 4.2 of the reference [24] from original paper [1] is included.

Lemma 2.
[24] Let G be any graph with vertices $v_{i}, 1 \leq i \leq p .$ Let the graphs $H_{i}$ with $n_{i}$ vertices, $1 \leq i \leq p,$ have $δ (H_{i}) \geq 1$ and each be antimagic. Further suppose that $Δ (H_{i}) \leq δ (H_{i + 1}), 1 \leq i \leq p - 1$ , and $Δ (H_{p}) \leq δ (G) .$ Then the compounding of $G,$ given by merging vertex $u_{j}^{i}$ in $H_{i}, 1 \leq i \leq p, 1 \leq j \leq n_{i},$ with the corresponding $v_{i}$ in $G,$ is antimagic.

Text correction: In the proof of Theorem 2, Case 2.2 is modified as follows:

Case 2.2. When

i < j

for

1 \leq i < j \leq n

. Consider,

ϕ_{h} (x) - ϕ_{h} (y) = ϕ_{f} (v_{i}) + ϕ_{g_{i}} (u_{p}^{i}) + k q (i - 1) + t n q - ϕ_{g_{j}} (u_{r}^{j}) - k q (j - 1)

We apply the maximum value for the negative terms and the minimum value for the positive terms in the above equations, i.e.,

j \leq n

and

i \geq 1,

ϕ_{h} (x) - ϕ_{h} (y) \geq ϕ_{f} (v_{i}) + ϕ_{g_{i}} (u_{p}^{i}) + t n q - ϕ_{g_{j}} (u_{r}^{j}) - k q (n - 1)

(10)

By the definition of f,

ϕ_{f} (v_{i}) \geq \frac{t (t + 1)}{2}

for any

i, 1 \leq i \leq n

, and by Lemma 1, we have the following:

ϕ_{g_{i}} (u_{p}^{i}) \leq k q - \frac{k (k - 1)}{2} and ϕ_{g_{j}} (u_{r}^{j}) \leq k q - \frac{k (k - 1)}{2}

By (10),

ϕ_{h} (x) - ϕ_{h} (y) \geq \frac{t (t + 1)}{2} + k q + (t - k) q n

(11)

If

t \geq k,

then

ϕ_{h} (x) \neq ϕ_{h} (y)

.

If

t < k

and if we let

t = k - c

for some constant

c \geq 1,

then Equation (11) is reduced into the following:

ϕ_{h} (x) - ϕ_{h} (y) = (k - c n) q + \frac{k (k - 2 c)}{2} + (\frac{k - c}{2}) + \frac{c^{2}}{2}

(12)

In particular, if

c = 1

we get,

ϕ_{h} (x) - ϕ_{h} (y) = (k - n) q + \frac{k (k - 2)}{2} + (\frac{k - 1}{2}) + \frac{1}{2}

(13)

By applying

k \geq n

in Equation (13), we obtain,

ϕ_{h} (x) - ϕ_{h} (y) > 0

.

Errors in Figures: In the original publication [1], there was a mistake in Figures 1–4 as published. As per the correction made in Theorem 1 the Figure 1, Figure 2, Figure 3 and Figure 4 along with its explanations is modified and appears below:

Figure 1, Figure 2, Figure 3 and Figure 4 illustrate the proof of Theorem 2. An antimagic labeling of a 2-regular graph G and the antimagic labeling of a 3-regular graph H are given in Figure 1 and Figure 2, respectively. In Figure 3, the three copies of the graph H are considered with their labeling function

g_{i}, 1 \leq i \leq 3

. The rooted product of the graph G and H with their antimagic labeling is given in Figure 4. Here, the root vertex of the graph H is chosen as

u_{6}

.

Figure 1. A graph G with

t = 2, n = 3, m = 3

.

Figure 1. A graph G with

t = 2, n = 3, m = 3

.

Figure 2. A graph H with

k = 3, p = 6, q = 9

.

Figure 2. A graph H with

k = 3, p = 6, q = 9

.

Figure 3. 3 copies of H.

Figure 4. Antimagic labeling of

G \circ_{v} H

.

Figure 4. Antimagic labeling of

G \circ_{v} H

.

Text included: A new paragraph is added after the illustration of Theorem 2 in which first paragraph gives the explanation and second paragraph gives the new construction.

Theorem

3.1

[24] is a generalized result for the antimagicn-ess of the corona product of graphs. It is worth mentioning that Theorem 3, which is given in our original manuscript [1], will be immediate according to Theorem

3.1

(cited [24] in [1]). Thus, the graph considered for

H

in Theorem 3 [1] is not true. In this correction, we prove Theorem 3 [1] with a new construction of the corona product by altering the conditions of generalized corona product is cited [24] in [1].

Let G be a connected graph of order n and size m, and let

H = \{H_{1}, H_{2}, \dots, H_{n}\}

with each

H_{i}, 1 \leq i \leq n

are connected regular graph of order

p_{i}

and size

q_{i}

.

The minimum and maximum degree of the graph G is denoted as

δ (G)

and

Δ (G)

. For

n = 3 l + r, 0 \leq r \leq 2

; then, consider

H = ⋃_{g = 1}^{l} P_{g} \cup \{H_{3 l + 1}, H_{3 l + 2}\}

, where

P_{g} = \{H_{3 g - 2}, H_{3 g - 1}, H_{3 g}\}

are called the

g^{t h}

pair of

H

. Then the following are true:

(i): For each pair $P_{g}, 1 \leq g \leq l, H_{3 g - 2}$ must be a complete graph of odd order and $p_{3 g - 2} \geq 4;$ $H_{3 g - 1}$ is a $P_{3 g - 2} - 2$ regular graph of order $p_{3 g - 2} + 1$ and $r e g (H_{3 g}) \geq p_{3 g - 2} .$
(ii): For any consecutive pairs, $P_{g}$ and $P_{g + 1}, g \geq 1$ , we must have $p_{3 g} < p_{3 (g + 1) - 2}$ and $r e g (H_{3 g}) < r e g (H_{3 (g + 1) - 2})$ .
Furthermore, if $r \neq 0$ , we have $p_{3 l} \leq p_{3 l + 1} \leq p_{3 l + 2}$ and $r e g (H_{3 l}) \leq r e g (H_{3 l + 1}) \leq r e g (H_{3 l + 2})$ .
(iii): The sum of the edges of the graphs in each pair $P_{g}, 1 \leq g \leq l$ is divisible by 2. Moreover, the sum of the edges in the graphs $\{H_{3 l + 1}, H_{3 l + 2}\}$ are also divisible by 2.
(iv): $δ (G) \geq r e g (H_{n}) - p_{1} + 1 .$

Note that,

x | 2

means, x is divisible by

2 .

Text correction: Theorem 3 and its proof is replaced with new theorem along with its proof according to the above construction.

Theorem 3.

Let G be a connected graph of order n and size m, and let

H

be a sequence of regular graphs under the conditions as defined above; then, the corona product of

G ⊙ H

is antimagic.

Proof.

Let G be a connected graph of order n and size

m .

Let

H = \{H_{1}, H_{2}, \dots, H_{n}\}

, with each

H_{i}, 1 \leq i \leq n

being a connected regular graph of order

p_{i}

and size

q_{i}

. Then, construct the corona graph by the above construction. We have

|E (G ⊙ H)| = \sum_{i = 1}^{n} (p_{i} + q_{i}) + m

. Now, we give the labels of the edges of

G ⊙ H

using the following steps:

First, we label the edges of

H_{i}, 1 \leq i \leq n

and then, we give labels to the edges that are incidents from the vertices of G to the vertices of

H_{i}

. Finally, we label the edges of G.

Case (i). If $n \equiv 0 (m o d 3) \Rightarrow n = 3 l,$ then $H = \cup_{g = 1}^{l} P_{g} .$ Label the edges of each pair $P_{g} = \{H_{3 g - 2}, H_{3 g - 1}, H_{3 g}\}$ as follows:

The edges of

H_{3 g - 2}

are labeled randomly from the set,

\{\sum_{s = 1}^{3 (g - 1)} q_{s} + 2 a - 1 : 1 \leq a \leq q_{3 g - 2}\} .

Then, the edges of

H_{3 g - 1}

are labeled randomly from the set,

\{\sum_{s = 1}^{3 (g - 1)} q_{s} + 2 a : 1 \leq a \leq q_{3 g - 1}\} .

The edges of

H_{3 g}

are labeled randomly from the set,

\{\sum_{s = 1}^{3 (g - 1)} q_{s} + 2 q_{3 g - 2} + a : 0 \leq a \leq q_{3 g} - 1\} .

Next, we label the edges of G randomly, from the set

\{\sum_{i = 1}^{n} (p_{i} + q_{i}) + a : 1 \leq a \leq m\}

.

Before labeling the remaining edges, we name the vertices of

H_{i}, 1 \leq i \leq n

and G based on the partial sum of their vertices, using the labeling defined above. The partial vertex sum for any vertex v is denoted as

ϕ^{'} (v)

and the vertex sum for any vertex v is denoted as

ϕ (v)

.

We name the vertices of

H_{i}, 1 \leq i \leq n

as

u_{i}^{1}, u_{i}^{2}, \dots, u_{i}^{p_{i}}

if

ϕ^{'} (u_{i}^{1}) \leq ϕ^{'} (u_{i}^{2}) \leq \dots \leq ϕ^{'} (u_{i}^{p_{i}})

.

We name the vertices of G as

v_{1}, v_{2}, \dots, v_{n}

if

ϕ^{'} (v_{1}) \leq ϕ^{'} (v_{2}) \leq \dots \leq ϕ^{'} (v_{n})

.

Now, we label the edges, incident to the vertex

u_{i}^{j}

of

H_{i}

and

v_{i}

of G, as

1 \leq i \leq n,

1 \leq j \leq p_{i}

, which is defined as follows:

For each pair

P_{g}

, the edge

u_{3 g - 2}^{j} v_{3 g - 2}

of

H_{3 g - 2}

for

1 \leq j \leq p_{3 g - 2}

is labelled as

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 (g - 1)} p_{s} + (2 j - 1) if \sum_{s = 1}^{3 (g - 1)} p_{s} | 2 . Otherwise, \sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 (g - 1)} p_{s} + 2 j .

The edge

u_{3 g - 1}^{j} v_{3 g - 1}

of

H_{3 g - 1}

for

1 \leq j \leq p_{3 g - 1}

is labelled as

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 (g - 1)} p_{s} + 2 j if \sum_{s = 1}^{3 (g - 1)} p_{s} | 2 . Otherwise, \sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 (g - 1)} p_{s} + 2 j - 1 .

The edge

u_{3 g}^{j} v_{3 g}

of

H_{3 g}

is labelled in the following way. If

\sum_{s = 1}^{3 (g - 1)} p_{s} | 2

, then the label of

u_{3 g}^{1} v_{3 g}

is

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 g - 1} p_{s}

and for

2 \leq j \leq p_{3 g}

, then the label of

u_{3 g}^{j} v_{3 g}

is

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 g - 1} p_{s} + j

.

If

\sum_{s = 1}^{3 (g - 1)} p_{s} ∤ 2

, then the edge

u_{3 g}^{j} v_{3 g}

is labelled as

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 g - 1} p_{s} + j

,

1 \leq j \leq p_{3 g}

.

Case (ii). If $n \equiv 1 (m o d 3) \Rightarrow n = 3 l + 1$ , then $H = \cup_{g = 1}^{l} P_{g} \cup \{H_{3 l + 1}\}$ .

First, we label the edges of

\cup_{g = 1}^{l} P_{g}

and G, and their incident edges, from

\cup_{g = 1}^{l} P_{g}

to G, as done in case (i).

Now, we label the edges of

H_{3 l + 1}

randomly from the set

\{\sum_{s = 1}^{3 l - 1} q_{s} + a : 1 \leq a \leq q_{3 l + 1}\}

. We name the vertices of

H_{3 l + 1}

using the partial sum as

u_{3 l + 1}^{1}, u_{3 l + 1}^{2}, u_{3 l + 1}^{3}, \dots, u_{3 l + 1}^{p_{3 l + 1}}

if

ϕ^{'} (u_{3 l + 1}^{1})

≤

ϕ^{'} (u_{3 l + 1}^{2}) \leq \dots \leq ϕ^{'} (u_{3 l + 1}^{p_{3 l + 1}})

.

Then, label the edge

u_{3 l + 1}^{j} v_{3 l + 1}

of

H_{3 l + 1}

as

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 g} p_{s} + j

for

1 \leq j \leq p_{3 l + 1}

.

Case (iii). If $n \equiv 2 (m o d 3) \Rightarrow n = 3 l + 2$ , then $H = \cup_{g = 1}^{l} P_{g} \cup \{H_{3 l + 1}, H_{3 l + 2}\}$ .

First, we label the edges of

\cup_{g = 1}^{l} P_{g}, H_{3 l + 1}

and G, and the incident edges, from

\cup_{g = 1}^{l} P_{g}, H_{3 l + 1}

to G, as done in case (ii).

We label the edges of

H_{3 l + 2}

randomly from the set

\{\sum_{s = 1}^{3 l + 1} q_{s} + a : 1 \leq a \leq q_{3 l + 2}\}

. Then, we name the vertices of

H_{3 l + 2}

using the partial vertex sum as

u_{3 l + 2}^{1}, u_{3 l + 2}^{2}, u_{3 l + 2}^{3}, \dots

,

u_{3 l + 2}^{p_{3 l + 2}}

if

ϕ^{'} (u_{3 l + 2}^{1}) \leq ϕ^{'} (u_{3 l + 2}^{2}) \leq \dots \leq ϕ^{'} (u_{3 l + 2}^{p_{3 l + 2}})

.

Then, we label the edge

u_{3 l + 2}^{j} v_{3 l + 2}

of

H_{3 l + 2}

as

\sum_{i = 1}^{n} q_{i} + \sum_{s = 1}^{3 g} p_{s} + p_{3 l + 1} + j

for

1 \leq j \leq p_{3 l + 2}

.

From the above labeling scheme, we can easily observe the following:

For each pair

P_{g}, P_{g} = \{H_{3 g - 2}, H_{3 g - 1}, H_{3 g}\}, 1 \leq g \leq l

we have,

Observation 2.

The vertex sum of

u_{3 g - 2}^{j}

,

ϕ (u_{3 g - 2}^{j})

is odd,

1 \leq j \leq p_{3 g - 2}

.

Observation 3.

The vertex sum of

u_{3 g - 1}^{j}

,

ϕ (u_{3 g - 1}^{j})

is even,

1 \leq j \leq p_{3 g - 1}

.

Claim: The above labeling of

G ⊙ H

is antimagic.

From the above labeling, we can easily observe that the vertex sum of the vertices of G satisfies

ϕ (v_{1}) < ϕ (v_{2}) < \dots < ϕ (v_{n})

. Therefore, any two vertices in G are different.

For any

i, 1 \leq i \leq n,

we observe that the vertices in

H_{i}

satisfy

ϕ (u_{i}^{1}) < ϕ (u_{i}^{2}) < \dots < ϕ (u_{i}^{p_{i}})

. From observation 2 and 3, we conclude that, for each pair

P_{g} = \{H_{3 g - 2}, H_{3 g - 1}, H_{3 g}\}

1 \leq g \leq l

, then

ϕ (u) \neq ϕ (v)

, where

u \in H_{3 g - 2}

and

v \in H_{3 g - 1}

. Also, we have

ϕ (u_{3 g}^{1}) > ϕ (u_{3 g - 2}^{p_{3 g - 2}})

.

ϕ (u_{3 g}^{1}) > ϕ (u_{3 g - 1}^{p_{3 g - 1}})

.

For any consecutive pairs

P_{g}

and

P_{g + 1}

, we have

ϕ (u_{3 g}^{p_{3 g}}) < ϕ (u_{3 (g + 1) - 2}^{1})

.

When

r \neq 0

, then we have

ϕ (u_{3 l}^{p_{3 l}}) < ϕ (u_{3 l + 1}^{1}) < ϕ (u_{3 l + 1}^{p_{3 l + 1}}) < ϕ (u_{3 l + 2}^{1}) < ϕ (u_{3 l + 2}^{p_{3 l + 2}})

.

Thus, any two vertices in

H

are distinct.

Now, we will prove that any vertex in

H

achieves a different vertex sum compared to any vertex sum in G. In order to prove this, we consider the minimum vertex sum in G and the maximum vertex sum in

H

. From the labeling, the vertex

v_{1}

gets the minimum sum in G, as follows:

ϕ (v_{1}) = [p_{1} + δ (G)] \sum_{i = 1}^{n} q_{i} + \sum_{i = 1}^{n} p_{i} δ (G) + n^{2} + \frac{δ (G) [δ (G) + 1]}{2}

(14)

and the vertex

u_{n}^{p_{n}}

gets the maximum vertex sum in

H

, as follows:

ϕ (u_{n}^{p_{n}}) = \sum_{i = 1}^{n} q_{i} [r e g (H_{n}) + 1] + \sum_{i = 1}^{n} p_{i} - \frac{r e g (H_{n}) [r e g (H_{n}) - 1]}{2}

(15)

Therefore, from Equations (14) and (15),

\begin{matrix} ϕ (v_{1}) - ϕ_{(} u_{n}^{p_{n}}) = [p_{1} + δ (G) - r e g (H_{n}) - 1] \sum_{i = 1}^{n} q_{i} + \sum_{i = 1}^{n} p_{i} [δ (G) - 1] + n^{2} + \\ \frac{δ (G) [δ (G) + 1]}{2} + \frac{r e g (H_{n}) [r e g (H_{n}) - 1]}{2} \end{matrix}

(16)

By applying

δ (G) \geq r e g (H_{n}) + 1 - p_{1}

to Equation (16), we obtain

ϕ (v_{1}) > ϕ_{f} (u_{n}^{p_{n}})

. Therefore, any two vertices in

G ⊙ H

receive distinct sums. □

Errors in Figures: In the original publication [1], the illustrations (Figures 5–8) of Theorem 3 is now been modified with the new illustrations based on the above Theorem 3.

A graph G shown in Figure 5 with their respective labelings as defined in the Theorem 3 and the graph

H_{1}, H_{2}, H_{3}, H_{4}

as shown in Figure 6, Figure 7, Figure 8 and Figure 9 with their respective labelings as defined in the Theorem 3. An antimagic labeling of graph

G ⊙ H

is shown in Figure 10.

Figure 5. The graph G.

Figure 6. The graph

H_{1}

having regularity 4,

p_{1} = 5, q_{1} = 10

.

Figure 6. The graph

H_{1}

having regularity 4,

p_{1} = 5, q_{1} = 10

.

Figure 7. The graph

H_{2}

having regularity 3,

p_{2} = 6, q_{2} = 9

.

Figure 7. The graph

H_{2}

having regularity 3,

p_{2} = 6, q_{2} = 9

.

Figure 8. The graph

H_{3}

having regularity 5,

p_{3} = 6, q_{3} = 15

.

Figure 8. The graph

H_{3}

having regularity 5,

p_{3} = 6, q_{3} = 15

.

Figure 9. The graph

H_{4}

having regularity 6,

p_{4} = 8, q_{4} = 34

.

Figure 9. The graph

H_{4}

having regularity 6,

p_{4} = 8, q_{4} = 34

.

Figure 10. The graph

G ⊙ H

.

Figure 10. The graph

G ⊙ H

.

Text Correction: In Section 3, Conclusions is modified and given below:

We proved an antimagic labeling for the rooted product of graphs G and H where G is a t-regular connected graph and H is a k-regular connected graph for

k \geq c n

for some

c \geq 1 .

Moreover, we proved that there exists an antimagic labeling for the corona product of regular graphs G and H under certain conditions.

In Author Contributions Section, Text correction:

Author Contributions: Conceptualization, M.V.; methodology, M.V.; investigation, V.L. and M.V.; Validation, M.V.; writing—original draft preparation, V.L. and M.V.; writing—review and editing, V.L. and M.V.; supervision, M.V.; M.V was an in charge and overall direction and planning of this research. All authors have read and agreed to the published version of the manuscript.

In Funding Section, Text correction:

Funding: The APC charge for this research is funded by Vellore Institute of Technology, Vellore, India.

The authors apologize sincerely for any inconvenience caused to the readers. With this correction, the order of some references has been adjusted accordingly. The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

References

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