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Factori-Difference Labeling to Some Line Graphs © 2018 JETIR September 2018, Volume 5, Issue 9 www.jetir.org (ISSN-2349-5162) FACTORI-DIFFERENCE LABELING TO SOME LINE GRAPHS 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] JETIR1809649 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 957 © 2018 JETIR September 2018, Volume 5, Issue 9 www.jetir.org (ISSN-2349-5162) 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. JETIR1809649 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 958 © 2018 JETIR September 2018, Volume 5, Issue 9 www.jetir.org (ISSN-2349-5162) 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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