Results and Discussion

VII. MINOR REVISION REQUIRED

2. Results and Discussion

Next theorem proves the exact value of the locating chromatic number for barbell graph𝐵_𝑛,𝑛.

Theorem 5. Let 𝐵_𝑛,𝑛 be a barbell graph for𝑛 ≥ 3. Then the locating chromatic number of𝐵_𝑛,𝑛is𝜒_𝐿(𝐵_𝑛,𝑛) = 𝑛 + 1.

Proof. Let𝐵_𝑛,𝑛,𝑛 ≥ 3, be the barbell graph with the vertex set𝑉(𝐵_𝑛,𝑛) = {𝑢_𝑖, V_𝑖 : 1 ≤ 𝑖 ≤ 𝑛} and the edge set 𝐸(𝐵_𝑛,𝑛)

=⋃^𝑛−1_𝑖=1{𝑢_𝑖𝑢_𝑖+𝑗 : 1 ≤ 𝑗 ≤ 𝑛 − 𝑖} ∪ ⋃^𝑛−1_𝑖=1{V_𝑖V_𝑖+𝑗 : 1 ≤ 𝑗 ≤ 𝑛 − 𝑖} ∪ {𝑢_𝑛V_𝑛}.

First, we determine the lower bound of the locating chromatic number for barbell graph 𝐵_𝑛,𝑛 for𝑛 ≥ 3. Since the barbell graph𝐵_𝑛,𝑛 contains two isomorphic copies of a complete graph𝐾_𝑛, then with respect to Theorem3we have 𝜒_𝐿(𝐵_𝑛,𝑛) ≥ 𝑛. Next, suppose that 𝑐 is a locating coloring using 𝑛 colors. It is easy to see that the barbell graph 𝐵_𝑛,𝑛

contains two vertices with the same color codes, which is a contradiction. Thus, we have that𝜒_𝐿(𝐵_𝑛,𝑛) ≥ 𝑛 + 1.

To show that 𝑛 + 1 is an upper bound for the locating chromatic number of barbell graph𝐵_𝑛,𝑛 it suffices to prove the existence of an optimal locating coloring𝑐 : 𝑉(𝐵_𝑛,𝑛) 󳨀→

{1, 2, . . . , 𝑛 + 1}. For 𝑛 ≥ 3 we construct the function 𝑐 in the following way:

𝑐 (𝑢_𝑖) = 𝑖, 1 ≤ 𝑖 ≤ 𝑛

𝑐 (V_𝑖) = {{ {{ {{ {{ {

𝑛, for𝑖 = 1 𝑖, for2 ≤ 𝑖 ≤ 𝑛 − 1 𝑛 + 1, otherwise.

(1)

By using the coloring𝑐, we obtain the color codes of 𝑉(𝐵_𝑛,𝑛) as follows:

𝑐_Π(𝑢_𝑖)

= {{ {{ {{ {{ {

0, for 𝑖^𝑡ℎ component, 1 ≤ 𝑖 ≤ 𝑛

2, for (𝑛 + 1)^𝑡ℎ component, 1 ≤ 𝑖 ≤ 𝑛 − 1 1, otherwise,

𝑐_Π(V_𝑖) = {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

0, for 𝑖^𝑡ℎ component, 2 ≤ 𝑖 ≤ 𝑛 − 1 for𝑛^𝑡ℎ component, 𝑖 = 1, and for (𝑛 + 1)^𝑡ℎ component, 𝑖 = 𝑛, 3, for 1^𝑠𝑡 component, 1 ≤ 𝑖 ≤ 𝑛 − 1 2, for 1^𝑠𝑡 component, 𝑖 = 𝑛 1, otherwise.

(2)

Since all vertices in𝑉(𝐵_𝑛,𝑛) have distinct color codes, then the coloring𝑐 is desired locating coloring. Thus, 𝜒_𝐿(𝐵_𝑛,𝑛) = 𝑛 + 1.

Corollary 6. For 𝑛, 𝑚 ≥ 3, and 𝑚 ̸= 𝑛, the locating chromatic number of barbell graph𝐵_𝑚,𝑛is

𝜒_𝐿(𝐵_𝑚,𝑛) = max {𝑚, 𝑛} . (3) Next theorem provides the exact value of the locating chromatic number for barbell graph𝐵_𝑃(𝑛,1).

Theorem 7. Let 𝐵_𝑃(𝑛,1)be a barbell graph for𝑛 ≥ 3. Then the locating chromatic number of𝐵_𝑃(𝑛,1)is

𝜒_𝐿(𝐵_𝑃(𝑛,1)) ={ {{

4, for odd 𝑛

5, for even 𝑛. (4)

Proof. Let𝐵_𝑃(𝑛,1),𝑛 ≥ 3, be the barbell graph with the vertex set𝑉(𝐵_𝑃(𝑛,1)) = {𝑢_𝑖, 𝑢_𝑛+𝑖, 𝑤_𝑖, 𝑤_𝑛+𝑖: 1 ≤ 𝑖 ≤ 𝑛} and the edge set 𝐸(𝐵_𝑃(𝑛,1)) = {𝑢_𝑖𝑢_𝑖+1, 𝑢_𝑛+𝑖𝑢_𝑛+𝑖+1, 𝑤_𝑖𝑤_𝑖+1,𝑤_𝑛+𝑖𝑤_𝑛+𝑖+1 : 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢_𝑛𝑢₁, 𝑢_2𝑛𝑢_𝑛+1, 𝑤_𝑛𝑤₁, 𝑤_2𝑛𝑤_𝑛+1} ∪ {𝑢_𝑖𝑢_𝑛+𝑖, 𝑤_𝑖𝑤_𝑛+𝑖: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢_𝑛𝑤_𝑛}.

Let us distinguish two cases.

International Journal of Mathematics and Mathematical Sciences 3 Case 1 (𝑛 odd). According to Theorem4for𝑛 odd we have

𝜒_𝐿(𝐵_𝑃(𝑛,1)) ≥ 4. To show that 4 is an upper bound for the locating chromatic number of the barbell graph𝐵_𝑃(𝑛,1) we describe an locating coloring𝑐 using 4 colors as follows:

𝑐 (𝑢_𝑖) = {{ {{ {{ {{ {

1, for 𝑖 = 1 3, for even 𝑖, 𝑖 ≥ 2 4, for odd 𝑖, 𝑖 ≥ 3.

𝑐 (𝑢_𝑛+𝑖) = {{ {{ {{ {{ {

2, for 𝑖 = 1 3, for odd 𝑖, 𝑖 ≥ 3 4, for even 𝑖, 𝑖 ≥ 2.

𝑐 (𝑤_𝑖) = {{ {{ {{ {{ {

1, for odd 𝑖, 𝑖 ≤ 𝑛 − 2 2, for even 𝑖, 𝑖 ≤ 𝑛 − 1 3, for 𝑖 = 𝑛.

𝑐 (𝑤_𝑛+𝑖) = {{ {{ {{ {{ {

1, for even 𝑖, 𝑖 ≤ 𝑛 − 1 2, for odd 𝑖, 𝑖 ≤ 𝑛 − 2 4, for 𝑖 = 𝑛.

(5)

For𝑛 odd the color codes of 𝑉(𝐵_𝑃(𝑛,1)) are

𝑐Π(𝑢𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for2^𝑛𝑑 component, 𝑖 ≤ 𝑛 + 1 2 𝑖 − 1, for1^𝑠𝑡component, 𝑖 ≤𝑛 + 1

2 𝑛 − 𝑖 + 1, for 1^𝑠𝑡component, 𝑖 >𝑛 + 1

2 𝑛 − 𝑖 + 2, for 2^𝑛𝑑 component, 𝑖 > 𝑛 + 1 0, for3^𝑡ℎ component, 𝑖 even, 𝑖 ≥ 22

for4^𝑡ℎ component, 𝑖 odd, 𝑖 ≥ 3 1, otherwise.

𝑐_Π(𝑢_𝑛+𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for1^𝑠𝑡 component, 𝑖 ≤𝑛 + 1 2 𝑖 − 1, for2^𝑛𝑑 component, 𝑖 ≤ 𝑛 + 1

2 𝑛 − 𝑖 + 1, for 2^𝑛𝑑 component, 𝑖 > 𝑛 + 1

2 𝑛 − 𝑖 + 2, for 1^𝑠𝑡 component, 𝑖 >𝑛 + 1 0, for4^𝑡ℎ component, 𝑖 even, 𝑖 ≥ 22

for3^𝑡ℎ component, 𝑖 odd, 𝑖 ≥ 3 1, otherwise.

𝑐Π(𝑤𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for3^𝑡ℎ component, 𝑖 ≤ 𝑛 − 1 2 𝑖 + 1, for4^𝑡ℎ component, 𝑖 ≤ 𝑛 − 1

2 𝑛 − 𝑖, for3^𝑡ℎ component, 𝑖 ≥ 𝑛 + 1

2 𝑛 − 𝑖 + 1, for 4^𝑡ℎ component, 𝑖 ≥ 𝑛 + 1 0, for2^𝑛𝑑 component, 𝑖 even, 𝑖 ≤ 𝑛 − 12

for1^𝑠𝑡 component, 𝑖 odd, 𝑖 ≤ 𝑛 − 2 1, otherwise.

𝑐_Π(𝑤_𝑛+𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for4^𝑡ℎ component, 𝑖 ≤ 𝑛 − 1 2 𝑖 + 1, for3^𝑡ℎ component, 𝑖 ≤ 𝑛 − 1

2 𝑛 − 𝑖, for4^𝑡ℎ component, 𝑖 ≥ 𝑛 + 1

2 𝑛 − 𝑖 + 1, for 3^𝑡ℎ component, 𝑖 ≥ 𝑛 + 1 0, for1^𝑠𝑡 component, 𝑖 even, 𝑖 ≤ 𝑛 − 12

for2^𝑛𝑑 component, 𝑖 odd, 𝑖 ≤ 𝑛 − 2 1, otherwise.

(6)

Since all vertices in𝐵_𝑃(𝑛,1)have distinct color codes, then the coloring𝑐 with 4 colors is an optimal locating coloring and it proves that𝜒_𝐿(𝐵_𝑃(𝑛,1)) ≤ 4.

Case 2 (𝑛 even). In view of the lower bound from Theorem7 it suffices to prove the existence of a locating coloring𝑐 : 𝑉(𝐵_𝑃(𝑛,1)) 󳨀→ {1, 2, . . . , 5} such that all vertices in 𝐵_𝑃(𝑛,1)

4 International Journal of Mathematics and Mathematical Sciences

have distinct color codes. For𝑛 even, 𝑛 ≥ 4, we describe the locating coloring in the following way:

𝑐 (𝑢_𝑖) = {{ {{ {{ {{ {{ {{ {{ {

1, for 𝑖 = 1

3, for even 𝑖, 2 ≤ 𝑖 ≤ 𝑛 − 2 4, for odd 𝑖, 3 ≤ 𝑖 ≤ 𝑛 − 1 5, for 𝑖 = 𝑛.

𝑐 (𝑢_𝑛+𝑖) = {{ {{ {{ {{ {

2, for 𝑖 = 1 3, for odd 𝑖, 𝑖 ≥ 3 4, for even 𝑖, 𝑖 ≥ 2.

𝑐 (𝑤_𝑖) = {{ {{ {{ {{ {{ {{ {{ {

1, for odd 𝑖, 𝑖 ≤ 𝑛 − 3 2, for even 𝑖, 𝑖 ≤ 𝑛 − 2 3, for 𝑖 = 𝑛 − 1 4, for 𝑖 = 𝑛.

𝑐 (𝑤_𝑛+𝑖) = {{ {{ {{ {{ {

1, for even 𝑖, 𝑖 ≤ 𝑛 − 2 2, for odd 𝑖, 𝑖 ≤ 𝑛 − 1 5, for 𝑖 = 𝑛.

(7)

In fact, our locating coloring of 𝐵_𝑃(𝑛,1), 𝑛 even, has been chosen in such a way that the color codes are

𝑐_Π(𝑢_𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for2^𝑛𝑑 and5^𝑡ℎcomponents, 𝑖 ≤ 𝑛 2 𝑖 − 1, for1^𝑠𝑡component, 𝑖 ≤𝑛

2 𝑛 − 𝑖, for5^𝑡ℎcomponent, 𝑖 > 𝑛 2 𝑛 − 𝑖 + 1, for 1^𝑠𝑡component, 𝑖 >𝑛 2 𝑛 − 𝑖 + 2, for 2^𝑛𝑑 component, 𝑖 >𝑛

0, for3^𝑡ℎcomponent, 𝑖 even, 2 ≤ 𝑖 ≤ 𝑛 − 22

for4^𝑡ℎcomponent, 𝑖 odd, 3 ≤ 𝑖 ≤ 𝑛 − 1 2, for4^𝑡ℎcomponent, 𝑖 = 1

for3^𝑡ℎcomponent, 𝑖 = 𝑛 1, otherwise.

𝑐_Π(𝑢_𝑛+𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for1^𝑠𝑡component, 𝑖 ≤𝑛 2 𝑖 − 1, for2^𝑛𝑑 component, 𝑖 ≤𝑛

2 𝑛 + 𝑖, for5^𝑡ℎcomponent, 𝑖 ≤ 𝑛 2

𝑛 − 𝑖 + 1, for 2^𝑛𝑑 and5^𝑡ℎcomponents, 𝑖 > 𝑛 2 𝑛 − 𝑖 + 2, for 1^𝑡ℎcomponent,𝑖 >𝑛

0, for3^𝑡ℎcomponent, 𝑖 odd, 3 ≤ 𝑖 ≤ 𝑛 − 12

for4^𝑡ℎcomponent, 𝑖 even, 2 ≤ 𝑖 ≤ 𝑛 2, for3^𝑡ℎcomponent, 𝑖 = 1

1, otherwise.

𝑐_Π(𝑤_𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for4^𝑡ℎcomponent, 𝑖 ≤ 𝑛 2 𝑖 + 1, for5^𝑡ℎcomponent, 𝑖 ≤ 𝑛 2 for3^𝑡ℎcomponent, 𝑖 ≤ 𝑛

2− 1 𝑛 − 𝑖, for4^𝑡ℎcomponent, 𝑖 > 𝑛

2 𝑛 − 𝑖 + 1, for 5^𝑡ℎcomponent, 𝑖 > 𝑛 2 𝑛 − 𝑖 − 1, for 3^𝑡ℎcomponent, 𝑛

2≤ 𝑖 ≤ 𝑛 − 1 0, for1^𝑠𝑡component, 𝑖 odd, 𝑖 ≤ 𝑛 − 3

for2^𝑛𝑑 component, 𝑖 even, 𝑖 ≤ 𝑛 − 2 2, for1^𝑠𝑡component, 𝑖 = 𝑛 − 1

for2^𝑛𝑑 component, 𝑖 = 𝑛 1, otherwise.

𝑐_Π(𝑤_𝑛+𝑖)

= {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑖, for5^𝑡ℎcomponent, 𝑖 ≤ 𝑛 2 𝑖 + 1, for4^𝑡ℎcomponent, 𝑖 ≤ 𝑛 2 𝑖 + 2 for3^𝑡ℎcomponent, 𝑖 ≤ 𝑛

2− 1 𝑛 − 𝑖, for3^𝑡ℎcomponent, 𝑛

2≤ 𝑖 ≤ 𝑛 − 1 for5^𝑡ℎcomponent, 𝑖 > 𝑛

2 𝑛 − 𝑖 + 1, for 4^𝑡ℎcomponent, 𝑖 > 𝑛

0, for1^𝑠𝑡component, 𝑖 even, 𝑖 ≤ 𝑛 − 22

for2^𝑛𝑑 component, 𝑖 odd, 𝑖 ≤ 𝑛 − 1 2, for1^𝑠𝑡and3^𝑡ℎcomponents, 𝑖 = 𝑛 1, otherwise.

(8)

Since for𝑛 even all vertices of 𝐵_𝑃(𝑛,1)have distinct color codes then our locating coloring has the required properties and 𝜒_𝐿(𝐵_𝑃(𝑛,1)) ≤ 5. This concludes the proof.

International Journal of Mathematics and Mathematical Sciences 5

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

3

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are thankful to DRPM Dikti for the Fundamental Grant 2018.

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