Factori-Difference Labeling to Some Line Graphs

Home , Cycle graph, Ladder graph

1A. Edward Samuel, 2S. Kalaivani 1Assistant Professor, 2Research Scholar 1, 2Ramanujan Research Centre, PG and Research Department of Mathematics Government Arts College (Autonomous), Kumbakonam – 612 001, Tamilnadu, India

Abstract : In this paper, we launch a new labeling said to be factori-difference labeling and apply to some line graphs. A connected graph G is a factori-difference labeling if there exists a bijection 푓 ∶ 푉(퐺) → {2, 3, … , 푝} such that the induced

[푓(푢)+푓(푣)−1]! function 푔 ∶ 퐸(퐺) → 푁 defined as 푔 (푢푣) = and the edges labels are distinct. Graph which acknowledges a 푓 푓 [푓(푢)−1]![푓(푣)−1]! factori-difference labeling is a factori-difference graph. We investigate this labeling conditions satisfies to some line graphs of path, cycle, brush, star, fan, friendship, ladder, sun let, triangular snake, dragon, key graphs and also find the chromatic number of some line graphs.

Keywords- Factori-difference labeling, Factori-difference graph, line graphs, Chromatic number.

I. INTRODUCTION Labeling of vertices of a graph such that the edges receive distinct labels has resulted in many interesting concepts. Beginning with graceful labeling in late 1960’s, this idea has emerged into a major branch of Graph Theory. Many graph theorists were attracted by this charming concept and have brought forth many varieties of labelings. Basic definitions are referred in Frank Harary[5]. A dynamic survey on graph labeling is systematic updated by Gallian[7] and it is published by Electronic Journal of Combinatorics. An enormous body of literature is available on different types of graph labeling grown around in the last four decades and more than 1000 research papers have been published. Graph labeling are using in many departments likely communication network, software testing, X-ray crystallography, etc., We investigate this labeling conditions satisfies to some line graphs of path, cycle, brush, star, fan, friendship, ladder, sun let, triangular snake, dragon, key graphs and also find the chromatic number of some line graphs.

II. PRELIMINARIES 2.1. Definition[8]

A walk in which no vertex is repeated is called a path 푃푛. It has 푛 vertices and 푛 − 1 edges.

2.2. Definition[8]

A closed path is called a cycle 퐶푛, 푛 ≥ 3. It has n vertices and n edges. 2.3. Definition[1]

The brush graph 퐵푛, (푛 ≥ 2) can be constructed by path graph 푃푛, (푛 ≥ 2) by joining the star graph 퐾1,1 at each vertex of the

path. i.e., 퐵푛 = 푃푛 + 푛퐾1,1. It has 2푛 vertices and 2푛 − 1 edges.

2.4. Definition[8]

A star graph 퐾1,푛 is said to be n pendant vertices with common central vertex. It has 푛 + 1 vertices and n edges.

2.5. Definition[12]

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A fan graph 퐹푛(푛 ≥ 2) is defined as the graph 퐾1 + 푃푛, where 퐾1 is the singleton graph and 푃푛 is the Path on n vertices. It has 푛 + 1 vertices and 2푛 − 1 edges.

2.6. Definition[11]

A cycle 퐶3 with n copies having a common central vertex is called a friendship graph 푇푛. It has 2푛 + 1 vertices and 3푛 edges.

2.7. Definition[2]

The cartesian product of path graphs 푃푛 x 푃2 is known as ladder graph 퐿푛, 푛 ≥ 2. It has 2푛 vertices and 3푛 − 2 edges.

2.8. Definition[9]

The cycle graph 퐶푛 with attaching n pendant vertices at each vertex is called the sun let graph 푆푛, 푛 ≥ 3. It has both 2푛 vertices and 2푛 edges.

2.9. Definition[3]

A Path graph 푃푛 by joining the vertices 푢푖 and 푢푖+1 to a new vertex 푣푗, 1 ≤ 푖, 푗 ≤ 푛 − 1 is said to be a triangular snake graph

푇푆푛, 푛 ≥ 2. That means every edge of the path is replaced by a triangle 퐶3. It has 2푛 − 1 vertices and 3푛 − 3 edges.

2.10. Definition[8]

A path 푃푚 is attached at one vertex of cycle 퐶푛 is called the dragon graph 퐷푛,푚, 푛 ≥ 3, 푚 ≥ 1. It has both 푛 + 푚 vertices and 푛 + 푚 edges.

2.11. Definition[10]

The key graph 퐾푌푚,푛, 푚 ≥ 3, 푛 ≥ 2 is a graph obtained from 퐾2 by appending one vertex of 퐶푚 to one end point and comb graph 푃푛⨀퐾1 to the other end of 퐾2. It has both (2푛 + 푚) vertices and (2푛 + 푚) edges.

2.12. Definition[4] A line graph L(G) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of G have a common vertex.

2.13. Definition[6] A coloring of a graph is an assigned color to its points so that two adjacent points have different colors and also non- adjacent vertices have either same color or any other colors. The chromatic number 휒(퐺) is defined as the minimum l for which a graph G has an l-coloring. III. FACTORI-DIFFERENCE LABELING TO SOME LINE GRAPHS

3.1. Definition A connected graph G is a factori-difference labeling if there exists a bijection 푓 ∶ 푉(퐺) → {2, 3, … , 푝} such that the

[푓(푢)+푓(푣)−1]! induced function 푔 ∶ 퐸(퐺) → 푁 defined as 푔 (푢푣) = and that edges labels are distinct. A graph which 푓 푓 [푓(푢)−1]![푓(푣)−1]! acknowledges a factori-difference labeling is a factori-difference graph.

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3.2. Theorem

The line graph 퐿(푃푛) of a path graph 푃푛, 푛 ≥ 2 is a factori-difference graph. Proof

Let 푃푛, (푛 ≥ 2) be the path graph with n vertices and 푛 − 1 edges. Let 푢1, 푢2, … , 푢푛 be the successive vertices of path graph and 푣푗 = 푢푖푢푖+1 for 1 ≤ 푖, 푗 ≤ 푛 − 1 be the edges of path 푃푛, (푛 ≥ 2). Let G be the line graph of a path graph 푃푛 is L(푃푛).

Here, 푉(퐺) = 푉(퐿(푃푛)) = {푣푗/1 ≤ 푗 ≤ 푛 − 1} and 퐸(퐺) = 퐸(퐿(푃푛)) = {푣푗푣푗+1 / 1 ≤ 푗 ≤ 푛 − 2}. Such that, |푉(퐿(푃푛))| = 푛 −

1 and |퐸(퐿(푃푛))| = 푛 − 2. The maximum degree is ∆= 2 and the minimum degree is 훿 = 1 of the line graph 퐿(푃푛). The bijection mapping is defined 푓 ∶ 푉(퐿(푃푛)) → {1, 2, … , 푛 − 1} as follows, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 푛 − 1. Then every edge 푒 =

[푓(푢)+푓(푣)−1]! 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗. Clearly, vertices and edges labels are distinct. Thus the function f is [푓(푢)−1]![푓(푣)−1]! 푖 푗 a factori-difference labeling for a line graph 퐿(푃푛). That is, the line graph 퐿(푃푛) of a path graph 푃푛 is a factori-difference graph.

The chromatic number 휒(퐿(푃푛)) of a line graph of path graph 퐿(푃푛) is the minimum l. i.e. 휒(퐿(푃푛)) is 2.

3.3. Example

The factori-difference labeling of line graph 퐿(푃푛), 푛 ≥ 2 of a path graph 푃푛 is shown figure 3.1.

(2)(퐺) (3)(퐵) (푛 − 1)(퐺) 푣1 (푛)(퐵) 푣 푣2 푣푛−2 푛−1

figure 3.1. factori-difference labeling for a graph 퐿(푃푛) and 휒(퐿(푃푛)) = 2.

3.4. Remark

The resultant graph of the line graph of n-path graph 퐿(푃푛) is a (푛 − 1)-path graph.

3.5. Theorem

The line graph 퐿(퐶푛) of a cycle graph 퐶푛, 푛 ≥ 3 acknowledges a factori-difference graph. Proof

Let the cycle graph 퐶푛, 푛 ≥ 3 be the both 푛 vertices and 푛 edges. Let 푢1, 푢2, … , 푢푛 be the successive vertices and 푣푗 =

푢푖푢푖+1 for 1 ≤ 푖, 푗 ≤ 푛 − 1 and 푣푛 = 푢푛푢1 be the edges of a cycle graph 퐶푛, 푛 ≥ 3. Let the resultant graph G be the line graph of a cycle graph and it is denoted by L(퐶푛), 푛 ≥ 3. Note that, 푉(퐺) = 푉(퐿(퐶푛)) = {푣푗/1 ≤ 푗 ≤ 푛} and 퐸(퐺) = 퐸(퐿(퐶푛)) =

{푣푗푣푗+1 / 1 ≤ 푗 ≤ 푛 − 1} ∪ {푣푛푣1}. Here, |푉(퐿(퐶푛))| = 푛 = |퐸(퐿(퐶푛))|. The maximum and minimum degree of line graph of a cycle graph 퐿(퐶푛) are both ∆= 2 = 훿. The function is defined 푓 ∶ 푉(퐿(퐶푛)) → {1, 2, … , 푛} as follows, 푓(푣푗) = 푗 + 1 for 1 ≤

[푓(푢)+푓(푣)−1]! 푗 ≤ 푛. Then the conditions are satisfied and that the conditions are every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ [푓(푢)−1]![푓(푣)−1]! 푖

푓(푒푗), 푖 ≠ 푗. Such that, every vertices and edges labels are distinct. Thus the function f is a factori-difference labeling for a line graph 퐿(퐶푛). Hence, the line graph 퐿(퐶푛) of a cycle graph 퐶푛 acknowledges a factori-difference graph. The chromatic number 2 if 푛 is even 휒(퐿(퐶 )) of a line graph of cycle graph 퐿(퐶 ) is the minimum l. i.e. 휒(퐿(퐶 )) = { . 푛 푛 푛 3 if 푛 is odd

3.6. Example

The factori-difference labeling of a line graph 퐿(퐶푛), 푛 ≥ 3 of a cycle graph 퐶푛 is shown figure 3.2.

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푣1(2)(퐺)

푣푛(푛 + 1)(퐵) 푣2(3)(퐵)

2 if 푛 is even figure 3.2. factori-difference labeling for a graph 퐿(퐶 ) and 휒(퐿(퐶 )) = { . 푛 푛 3 if 푛 is odd

3.7. Remark

The resultant graph of the line graph of a cycle graph 퐿(퐶푛) is also a cycle graph 퐶푛. i.e., 퐿(퐶푛) ≅ 퐶푛. 3.8. Theorem

The line graph of a brush graph 퐿(퐵푛), 푛 ≥ 2 admits a factori-difference graph. Proof

Let the brush graph 퐵푛, 푛 ≥ 2 has 2푛 vertices and 2푛 − 1 edges. Let G be the resultant graph of a line graph of brush graph

퐿(퐵푛), 푛 ≥ 2 has 2푛 − 1 vertices and 3푛 − 4 edges. Let 푉(퐺) = 푉(퐿(퐵푛)) = {푣푗 / 1 ≤ 푗 ≤ 2푛 − 1} and 퐸(퐺) = 퐸(퐿(퐵푛)) =

{푣푗푣푗+1 / 1 ≤ 푗 ≤ 푛 − 1} ∪ {푣푗푣2푛−푗 / 1 ≤ 푗 ≤ 푛 + 1} ∪ {푣푗푣(2푛−1)−푗 / 1 ≤ 푗 ≤ 푛}. Here, |푉(퐿(퐵푛))| = 2푛 − 1 and

|퐸(퐿(퐵푛))| = 3푛 − 4. The maximum degree ∆= 4 and the minimum degree 훿 = 1 of a line graph of brush graph 퐿(퐵푛). The function mapping 푓: 푉(퐿(퐵푛)) → {1, 2, … , 2푛 − 1} is defined as, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 2푛 − 1. i.e., the factori-difference

[푓(푢)+푓(푣)−1]! labeling conditions are every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗. So that, vertices and edges [푓(푢)−1]![푓(푣)−1]! 푖 푗 labels are distinct. Hence, f is a factori-difference labeling. Therefore, 퐿(퐵푛), 푛 ≥ 2 admits a factori-difference graph. The chromatic number of a line graph of brush graph, 휒(퐿(퐵푛)) = 3. 3.9. Example

The factori-difference labeling of a line graph 퐿(퐵푛), 푛 ≥ 2 of a brush graph 퐵푛 is shown figure 3.3.

푣푛+1(푛 + 2)(퐺) 푣2푛−2(2푛 − 1)(퐺) 푣2푛−3(2푛 − 2)(퐺) 푣푛(푛 + 1)(퐺) 푣2푛−1(2푛)(퐺)

푣1(2)(퐵) 푣 (3)(푌) 푣 (푛)(퐵) 2 푛−1

figure 3.3. factori-difference labeling for a graph 퐿(퐵푛), 푛 ≥ 2 and 휒(퐿(퐵푛)) = 3. 3.10. Remark

The resultant graph of the line graph of a brush graph 퐿(퐵푛) is the middle graph of a path graph 푀(푃푛).

3.11. Theorem

The line graph of a star graph 퐿(퐾1,푛)has a factori-difference graph. Proof

Let the star graph 퐾1,푛 has 푛 + 1 vertices and 푛 edges. i.e., the star graph has one common vertex and attached 푛 pendant 푛(푛−1) vertices. Let G be the resultant graph of a line graph of star graph 퐿(퐾 ) has 푛 vertices and edges. Let 푉(퐺) = 1,푛 2 푛(푛−1) 푉(퐿(퐾 )) = {푣 / 1 ≤ 푗 ≤ 푛}. Note that, |푉(퐿(퐾 ))| = 푛 and |퐸(퐿(퐾 ))| = . The maximum and minimum degree of a 1,푛 푗 1,푛 1,푛 2 line graph of star graph 퐿(퐾1,푛) are same. i.e., ∆= 푛 − 1 = 훿. The mapping 푓: 푉(퐿(퐾1,푛)) → {1, 2, … , 푛} is defined by, 푓(푣푗) =

[푓(푢)+푓(푣)−1]! 푗 + 1 for 1 ≤ 푗 ≤ 푛. Such that, the factori-difference labeling conditions i.e., every edge 푒 = 푢푣 = and for any [푓(푢)−1]![푓(푣)−1]!

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edge 푓(푒푖) ≠ 푓(푒푗), 푖 ≠ 푗 are satisfied. So that, vertices and edges labels are distinct. Hence, f is a factori-difference labeling and also 퐿(퐾1,푛) has a factori-difference graph. The chromatic number of a line graph of star graph, 휒(퐿(퐾1,푛)) = 푛.

3.12. Example

The factori-difference labeling of a line graph 퐿(퐾1,푛)of a star graph 퐾1,푛 is shown figure 3.4.

(2)(퐺) (3)(퐵) (푛)(푌) (푛 + 1)(푃) 푣1 푣 푣 푣 2 푛−1 푛

figure 3.4. factori-difference labeling for a graph 퐿(퐾1,푛)and 휒 (퐿(퐾1,푛)) = 푛.

3.13. Remark 푛−1 The resultant graph of the line graph of a star graph 퐿(퐾1,푛) is a complete graph or (푛 − 1)- power path graph (푃 ).

3.14. Theorem

The line graph 퐿(퐹푛), 푛 ≥ 2 of a fan graph 퐹푛, 푛 ≥ 2 acknowledges a factori-difference graph. Proof

Let the successive vertices of fan graph 퐹푛, 푛 ≥ 2 be 푢1, 푢2, … , 푢푛+1. i.e., 푛 + 1 vertices and 2푛 − 1 edges. Let G be the 푛2+5푛−8 line graph of a fan graph L(퐹 ), 푛 ≥ 2 with 2푛 − 1 vertices and edges. i.e., 푉(퐺) = 푉(퐿(퐹 )) = {푣 /1 ≤ 푗 ≤ 2푛 − 1}. 푛 2 푛 푗 푛2+5푛−8 2, 푛 = 2 Here, |푉(퐿(퐹 ))| = 2푛 − 1 and |퐸(퐿(퐹 ))| = . The maximum degree of a line graph 퐿(퐹 ) is ∆= { and the 푛 푛 2 푛 푛 + 1, 푛 ≥ 3 2, 푛 = 2 minimum degree of a line graph 퐿(퐹 ) is 훿 = { . The function is defined 푓 ∶ 푉(퐿(퐹 )) → {1, 2, … , 2푛 − 1} by, 푓(푣 ) = 푛 3, 푛 ≥ 3 푛 푗 [푓(푢)+푓(푣)−1]! 푗 + 1 for 1 ≤ 푗 ≤ 2푛 − 1. Then, for every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗. Clearly, [푓(푢)−1]![푓(푣)−1]! 푖 푗 vertices and edges labels are distinct. Thus the function f satisfied the factori-difference labeling conditions. Hence, the line graph 퐿(퐹푛), 푛 ≥ 2 of a fan graph 퐹푛, 푛 ≥ 2 acknowledges a factori-difference graph. The chromatic number 휒(퐿(퐹푛)) of a 3, 푛 = 2 line graph of fan graph 퐿(퐹 ) is the minimum l. i.e. 휒(퐿(퐹 )) = { . 푛 푛 푛, 푛 ≥ 3

3.15. Example

The factori-difference labeling of line graph 퐿(퐹푛), 푛 ≥ 2 of a fan graph 퐹푛, 푛 ≥ 2 is shown figure 3.5.

푣2푛−2(2푛 − 1)(푃)

푣푛+3(푛 + 4)(퐵) 푣2푛−1(2푛)(푂) 푣푛+2(푛 + 3)(푌)

푣1(2)(퐵) 푣2(3)(푌) 푣푛(푛 + 1)(푂) 푣푛+1(푛 + 2)(퐺)

3 if 푛 = 2 figure 3.5. factori-difference labeling for a graph 퐿(퐹 ), 푛 ≥ 2 and 휒(퐿(퐹 )) = { . 푛 푛 푛 if 푛 ≥ 3

3.16. Theorem

The line graph of a friendship graph 퐿(푇푛) has a factori-difference graph.

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Proof

Let 푇푛 be a friendship graph with 2푛 + 1 vertices and 3푛 edges. i.e., 푉(푇푛) = {푢1, 푢2, … , 푢2푛+1} with 푢1 as a common vertex and 퐸(푇푛) = {푢1푢푖 / 2 ≤ 푖 ≤ 2푛 + 1} ∪ {푢2푖푢2푖+1 / 1 ≤ 푖 ≤ 푛}. Let G be the resultant graph of a line graph of friendship 2 graph 퐿(푇푛) with 3푛 vertices and 2푛 + 푛 edges. i.e., 푉(퐺) = 푉(퐿(푇푛)) = {푣푗 / 1 ≤ 푗 ≤ 3푛}. Note that, |푉(퐿(푇푛))| = 3, 푛 = 1 3푛 and |퐸(퐿(푇 ))| = 2푛2 + 푛 . The maximum and minimum degree of a line graph of friendship graph 퐿(푇 ) are ∆= { 푛 푛 2푛, 푛 ≥ 2 and 훿 = 2 respectively. The bijecting mapping function 푓: 푉(퐿(푇푛)) → {1, 2, … , 3푛} is defined as, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ [푓(푢)+푓(푣)−1]! 3푛. Such that, the factori-difference labeling conditions i.e., every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ [푓(푢)−1]![푓(푣)−1]! 푖

푓(푒푗), 푖 ≠ 푗 are satisfied. So that, vertices and edges labels are distinct. Hence, f is a factori-difference labeling and also 퐿(푇푛) 3, 푛 = 1 has a factori-difference graph. The chromatic number of a line graph of friendship graph 퐿(푇 ),휒(퐿(푇 )) = { . 푛 푛 2푛, 푛 ≥ 2 3.17. Example

The factori-difference labeling of a line graph 퐿(푇푛) of a friendship graph 푇푛 is shown figure 3.6.

푣3푛(3푛 + 1)(푃) 푣2푛+1(2푛 + 2)(퐵)

(2)(퐺) (3)(퐵) (2푛)(푌) 푣1

푣2 푣 푣 (2푛 + 1)(푃) 2푛−1 2푛 3 if 푛 = 1 figure 3.6. factori-difference labeling for a graph 퐿(푇 ) and 휒(퐿(푇 )) = { . 푛 푛 2푛 if 푛 ≥ 2

3.18. Theorem

The line graph of a ladder graph 퐿(퐿퐷푛), 푛 ≥ 2 produces a factori-difference graph. Proof

Let the ladder graph 퐿퐷푛, 푛 ≥ 2 with 2푛 vertices and 3푛 − 2 edges. i.e., 푉(퐿퐷푛) = 2푛 and 퐸(퐿퐷푛) = 3푛 − 2. Let

푢1, 푢2, … , 푢2푛 be the successive vertices of ladder graph 퐿퐷푛, 푛 ≥ 2 and the edge set is 퐸(퐿퐷푛) = {푢푖푢푖+1 / 1 ≤ 푖 ≤ 2푛 − 1} ∪

{푢푖푢2푛−푖+1 /1 ≤ 푖 ≤ 푛}. Let G be the resultant graph of a line graph of ladder graph 퐿(퐿퐷푛), 푛 ≥ 2 with 3푛 − 2 vertices and

6푛 − 8 edges. Here, 푉(퐺) = 푉(퐿(퐿퐷푛)) = {푣푗 / 1 ≤ 푗 ≤ 3푛 − 2} and 퐸(퐺) = 퐸(퐿(퐿퐷푛)) = {푣푗푣푗+1 / 1 ≤ 푗 ≤ 2푛 − 1} ∪

{푣푗푣2푛−1+푗 / 1 ≤ 푗 ≤ 푛 − 1} ∪ {푣푗푣2푛+푗 / 1 ≤ 푗 ≤ 푛 − 2} ∪ {푣푗푣4푛−1−푗 / 푛 + 1 ≤ 푗 ≤ 2푛 − 1} ∪ {푣푗푣4푛−푗 / 푛 + 2 ≤ 푗 ≤ 2푛 −

1}. i.e., |푉(퐿(퐿퐷푛))| = 3푛 − 2 and |퐸(퐿(퐿퐷푛))| = 6푛 − 8. The maximum and minimum degree of a line graph of ladder graph

퐿(퐿퐷푛) are ∆= 4 and 훿 = 2 respectively. The mapping 푓: 푉(퐿(퐿퐷푛)) → {1, 2, … , 3푛 − 2} is defined as, 푓(푣푗) = 푗 + 1 for 1 ≤ [푓(푢)+푓(푣)−1]! 푗 ≤ 3푛 − 2. Such that, the factori-difference labeling conditions i.e., every edge 푒 = 푢푣 = and for any edge [푓(푢)−1]![푓(푣)−1]!

푓(푒푖) ≠ 푓(푒푗), 푖 ≠ 푗 are satisfied. Then, the vertices and edges labels are distinct. Hence, f is a factori-difference labeling and also 퐿(퐿퐷푛) produces a factori-difference graph. The chromatic number of a line graph of ladder graph 퐿(퐿퐷푛), 휒(퐿(퐿퐷푛)) = 3.

3.19. Example

The factori-difference labeling of a line graph 퐿(퐿퐷푛), 푛 ≥ 2 of a ladder graph 퐿퐷푛 is shown figure 3.7.

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푣 (푛 − 1)(퐵) 푣2(3)(푌) 푛−2 푣 (푛)(푌) 푣1(2)(퐵) 푛−1

푣3푛−2(3푛 − 1)(퐺) 푣2푛+1(2푛 + 2)(퐺) 푣푛(푛 + 1)(퐵) 푣2푛(2푛 + 1)(푌)

푣2푛−1(2푛)(퐵) 푣푛+1(푛 + 2)(푌) 푣푛+2(푛 + 3)(퐵) 푣 (2푛 − 1)(푌) 2푛−2

figure 3.7. factori-difference labeling for a graph 퐿(퐿퐷푛)and

휒(퐿(퐿퐷푛)) = 3.

3.20. Theorem

The line graph of a sun let graph 퐿(푆푛), 푛 ≥ 3 is a factori-difference graph. Proof

Let 푆푛, 푛 ≥ 3 be the sun let graph with 2푛 vertices and 2푛 edges. Let the vertex and edge set of sun let graph are 푉(푆푛) =

{푢푖 / 1 ≤ 푖 ≤ 2푛} and 퐸(푆푛) = {푢푖푢푖+1 / 1 ≤ 푖 ≤ 푛 − 1} ∪ {푢푛푢1} ∪ {푢푖푢푛+푖 /1 ≤ 푖 ≤ 푛}. Let G be the resultant graph of a line graph of sun let graph 퐿(푆푛), 푛 ≥ 3 with 2푛 vertices and 3푛 edges. Here, 푉(퐺) = 푉(퐿(푆푛)) = {푣푗 / 1 ≤ 푗 ≤ 2푛} and 퐸(퐺) =

퐸(퐿(푆푛)) = {푣푗푣푗+1 / 1 ≤ 푗 ≤ 푛} ∪ {푣푛푣1} ∪ {푣푗푣푛+푗 / 1 ≤ 푗 ≤ 푛} ∪ {푣푗푣푛+1+푗 / 1 ≤ 푗 ≤ 푛 − 1}. i.e., |푉(퐿(푆푛))| = 2푛 and

|퐸(퐿(푆푛))| = 3푛. The maximum and minimum degree of a line graph of sun let graph 퐿(푆푛) are ∆= 4 and 훿 = 2 respectively.

The labeling function is 푓: 푉(퐿(푆푛)) → {1, 2, … , 2푛} defined by, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 2푛. Then, for every edge 푒 = 푢푣 =

[푓(푢)+푓(푣)−1]! and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗. Such that, the vertices and edges labels are distinct. So, the factori- [푓(푢)−1]![푓(푣)−1]! 푖 푗 difference labeling conditions are satisfied. Hence, f is a factori-difference labeling and also 퐿(푆푛), 푛 ≥ 3 is a factori-difference graph. The chromatic number of a line graph of sun let graph 퐿(푆푛), 휒(퐿(푆푛)) = 3.

3.21. Example

The factori-difference labeling of a line graph 퐿(푆푛), 푛 ≥ 3 of a sun let graph 푆푛 is shown figure 3.8.

푣1(2)(퐺) 푣 (푛 + 3)(푌) 푣푛+1(푛 + 2)(푌) 푛+2

푣2(3)(퐵) 푣푛(푛 + 1)(퐵)

( ) 푣 (2푛 + 1)(푌) 푣푛−1 푛 (퐺) 2푛

figure 3.8. factori-difference labeling for a graph 퐿(푆푛) and 휒(퐿(푆푛)) = 3.

3.22. Theorem

The line graph of a triangular snake graph 퐿(푇푆푛), 푛 ≥ 2 admits a factori-difference graph. Proof

Let 푇푆푛, 푛 ≥ 2 be the triangular snake graph with 2푛 − 1 vertices and 3푛 − 3 edges. Here, the successive vertices of triangular snake graph 푇푆푛, 푛 ≥ 2 are 푢1, 푢2, … , 푢2푛−1. Let G be the resultant graph of a line graph of triangular snake graph

퐿(푇푆푛), 푛 ≥ 2 with 3푛 − 3 vertices and 7푛 − 11 edges. i.e., 푉(퐺) = 푉(퐿(푇푆푛)) = {푣푗 / 1 ≤ 푗 ≤ 3푛 − 3} and

퐸(퐺) = 퐸(퐿(푇푆푛)) = {푣푗푣푗+1 / 1 ≤ 푗 ≤ 3푛 − 4} ∪ {푣푗푣3푛−4−푗 / 1 ≤ 푗 ≤ 푛 − 2} ∪ {푣푗푣3푛−3−푗 / 1 ≤ 푗 ≤ 푛 − 3} ∪

{푣푗푣3푛−2−푗 / 1 ≤ 푗 ≤ 푛 − 4} ∪ {푣푗푣2푛−푗 / 3 ≤ 푗 ≤ 푛 − 3} ∪ {푣푗푣2푛+1−푗 / 푛 − 4 ≤ 푗 ≤ 푛 − 1} ∪ {푣푗푣2푛−1−푗, 푣푗푣2푛+2−푗 / 푗 =

푛 − 2} ∪ {푣푗푣2푛−푗, 푣푗푣2푛+3−푗 / 푗 = 푛 − 3}. Note that, |푉(퐿(푇푆푛))| = 3푛 − 3 and |퐸(퐿(푇푆푛))| = 7푛 − 11. The maximum and

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minimum degree of a line graph of a triangular snake graph 퐿(푇푆푛) are ∆= 6 and 훿 = 2 respectively. The labeling function

푓: 푉(퐿(푇푆푛)) → {1, 2, … , 3푛 − 3} is defined as, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 3푛 − 3. Such that, the factori-difference labeling [푓(푢)+푓(푣)−1]! conditions for every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗 are satisfied. Clearly, the vertices and [푓(푢)−1]![푓(푣)−1]! 푖 푗 edges labels are distinct. Hence, f is a factori-difference labeling and therefore 퐿(푇푆푛) admits a factori-difference graph. The chromatic number of a line graph of a triangular snake graph 퐿(푇푆푛), 휒(퐿(푇푆푛)) = 4.

3.23. Example

The factori-difference labeling of a line graph 퐿(푇푆푛), 푛 ≥ 2 of the triangular snake graph 푇푆푛 is shown figure 3.9.

푣3푛−4(3푛 − 3)(퐺) 푣푛+1(푛 + 2)(퐺) 푣 (푛 + 1)(퐵) 푣 (3푛 − 2)(푌) 푛 3푛−3 푣3푛−5(3푛 − 4)(푃)

푣푛−1(푛)(푌) 푣3(4)(퐵) 푣 (푛 − 1)(퐵) 푣 (2)(퐵) 푣2(3)(푌) 푛−2 1

figure 3.9. factori-difference labeling for a graph 퐿(푇푆푛) and 휒(퐿(푇푆푛)) = 4.

3.24. Theorem

The line graph of a dragon graph 퐿(퐷푛,푚), 푛 ≥ 3, 푚 ≥ 1 has a factori-difference graph. Proof

Let 퐷푛,푚, 푛 ≥ 3, 푚 ≥ 1 be a dragon graph with both 푛 + 푚 vertices and 푛 + 푚 edges. Let

푢1, 푢2, … , 푢푛, 푢푛+1, 푢푛+2, … , 푢푛+푚 be the successive vertices of a dragon graph 퐷푛,푚, 푛 ≥ 3, 푚 ≥ 1. Let 퐺 = 퐿(퐷푛,푚), 푛 ≥

3, 푚 ≥ 1 be the resultant graph of a line graph of dragon graph with 푣1, 푣2, … , 푣푛, 푣푛+1, 푣푛+2, … , 푣푛+푚 be the successive vertices of a line graph of the dragon graph. Let the line graph of a dragon graph be 푛 + 푚 vertices and 푛 + 푚 + 1 edges. Such that,

푉(퐺) = 푉(퐿(퐷푛,푚)) = {푣푗 / 1 ≤ 푗 ≤ 푛 + 푚} and 퐸(퐺) = 퐸 (퐿(퐷푛,푚)) = {푣푗푣푗+1 / 1 ≤ 푗 ≤ 푛 + 푚 − 1} ∪ {푣푛푣1} ∪ {푣푛+1푣1}. i.e., |푉(퐿(퐷푛,푚))| = 푛 + 푚 and |퐸(퐿(퐷푛,푚))| = 푛 + 푚 + 1. The maximum degree of a line graph of dragon graph 퐿(퐷푛,푚) is

∆= 3 and the minimum degree of a line graph of dragon graph 퐿(퐷푛,푚) is 훿 = 1. The labeling mapping function is defined

푓: 푉(퐿(퐷푛,푚)) → {1, 2, … , 푛 + 푚} by, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 푛 + 푚. That is, the factori-difference labeling conditions for [푓(푢)+푓(푣)−1]! every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗 are satisfied. Then, the vertices and edges labels are [푓(푢)−1]![푓(푣)−1]! 푖 푗 distinct. Hence, the labeling mapping function f is a factori-difference labeling and the line graph of a dragon graph 퐿(퐷푛,푚) has a factori-difference graph. The chromatic number of a line graph of a dragon graph 퐿(퐷푛,푚), 휒(퐿(퐷푛,푚)) = 3.

3.25. Example

The factori-difference labeling of a line graph 퐿(퐷푛,푚), 푛 ≥ 3, 푚 ≥ 1 of a dragon graph 퐷푛,푚 is shown figure 3.10.

푣푛(푛 + 1)(푌) 푣2(3)(푌)

푣푛+푚−1(푛 + 푚)(퐵) 푣1(2)(퐵) 푣 (푛 + 2)(퐺) 푢푛+푚(푛 + 푚 + 1)(푌) 푛+1

figure 3.10. factori-difference labeling for a graph 퐿(퐷푛,푚) and 휒 (퐿(퐷푛,푚)) = 3.

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3.26. Theorem

The line graph of a key graph 퐿(퐾푌푚,푛), 푚 ≥ 3, 푛 ≥ 2 acknowledges a factori-difference graph. Proof

Let 푢1, 푢2, … , 푢푚, 푢푚+1, 푢푚+2, … , 푢2푛+푚 be the successive vertices of a key graph 퐾푌푚,푛, 푚 ≥ 3, 푛 ≥ 2. Here, the key

graph 퐾푌푚,푛, 푚 ≥ 3, 푛 ≥ 2 with both 2푛 + 푚 vertices and 2푛 + 푚 edges. Let 퐺 = 퐿(퐾푌푚,푛), 푚 ≥ 3, 푛 ≥ 2 be the resultant graph

of a line graph of a key graph with 푣1, 푣2, … , 푣푚, 푣푚+1, 푣푚+2, … , 푣2푛+푚 be the successive vertices of a line graph of a key graph.

i.e., 2푛 + 푚 vertices and 3푛 + 푚 edges. Note that, 푉(퐺) = 푉(퐿(퐾푌푚,푛)) = {푣푗 / 1 ≤ 푗 ≤ 2푛 + 푚} and 퐸(퐺) = 퐸 (퐿(퐾푌푚,푛)) =

{푣푗푣푗+1 / 1 ≤ 푗 ≤ 푚 + 푛} ∪ {푣푚푣1} ∪ {푣2푛푣1} ∪ {푣푗푣2(푛+푚)+1−푗 / 푚 + 1 ≤ 푗 ≤ 푚 + 푛} ∪ {푣푗푣2(푛+푚)+2−푗 / 푚 + 2 ≤ 푗 ≤ 푚 +

푛}. i.e., |푉(퐿(퐾푌푚,푛))| = 2푛 + 푚 and |퐸(퐿(퐾푌푚,푛))| = 3푛 + 푚. The maximum degree of a line graph of a key graph 퐿(퐾푌푚,푛)

is ∆= 4 and the minimum degree of a line graph of a key graph 퐿(퐾푌푚,푛) is 훿 = 2. The labeling function is defined

푓: 푉(퐿(퐾푌푚,푛)) → {1, 2, … , 2푛 + 푚} by, 푓(푣푗) = 푗 + 1 for 1 ≤ 푗 ≤ 2푛 + 푚. Then, the factori-difference labeling conditions [푓(푢)+푓(푣)−1]! for every edge 푒 = 푢푣 = and for any edge 푓(푒 ) ≠ 푓(푒 ), 푖 ≠ 푗 are satisfied. i.e., the vertices and edges labels are [푓(푢)−1]![푓(푣)−1]! 푖 푗

distinct. Thus, the labeling function f is a factori-difference labeling and the line graph of a key graph 퐿(퐾푌푚,푛), 푚 ≥ 3, 푛 ≥ 2

acknowledges a factori-difference graph. The chromatic number of a line graph of a key graph 퐿(퐾푌푚,푛), 휒(퐿(퐾푌푚,푛)) = 3.

3.27. Example

The factori-difference labeling of a line graph 퐿(퐾푌푚,푛), 푚 ≥ 3, 푛 ≥ 2 of a key graph 퐾푌푚,푛 is shown figure 3.11.

푣푚+1(푚 + 2)(퐺) 푣푚+푛−1(푚 + 푛)(퐵) 푣푚(푚 + 1)(푌) 푣푚+푛(푚 + 푛 + 1)(푌) 푣푚+2(푚 + 3)(푌) 푣푚−1(푚)(퐵) 푣1(2)(퐵) 푣2푛+푚(2푛 + 푚 + 1)(퐵) 푣푚+푛+1(푚 + 푛 + 2)(퐵)

figure 3.11. factori-difference labeling for a graph 퐿(퐾푌푚,푛) and 휒 (퐿(퐾푌푚,푛)) = 3. IV. ACKNOWLEDGMENT The author is thankful to the referee for the valuable comments in this paper.

V. CONCLUSION In this paper, we investigated the factori-difference labeling and that the factori-difference graph. The factori-difference labeling conditions are satisfied to some line graphs of a classes of graphs namely path, cycle, brush, star, fan, friendship, ladder, sun let, triangular snake, dragon, key graphs and also the above graphs are admits the factori-difference graphs. Also, we found that the chromatic number of the line graph of some classes of graphs.

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