Equality Labeling on Special Graphs

EQUALITY LABELING ON SPECIAL GRAPHS

Dr.A.MYDEEN BIBI1 AND M.MALATHI2 1Assistant Professor of Mathematics, the Standard Firework Rajaratnam College for Women, Sivakasi 2Research Scholar the Standard Firework Rajaratnam College for Women, Sivakasi Abstract- Let G be a graph of order n and size m. A onto function : v (G) → {0,1,2,...,n} is called a equality labeling. If it satisfies following condition  The edges labeled by {1,2,3,…,2m-1} with │f(u)+f(v)│ if n=m.  The edges labeled by{1,2,3,…,m} with │f(u)-f(v)│if n≠m. This labeling is called an equality labeling. It is denoted by Eq (G). In this paper we investigate the behavior of equality labeling for some special graphs. Keywords- Equality labeling

I. INTRODUCTION

If G is a graph, let V (G) and E (G) denote, respectively, the vertex and the edge set of G. We deal with labeling with domain either the set of all vertices, or the set of all edges, or the set of all vertices and edges, respectively. We call these labeling a vertex labeling, or an edge labeling, or a total labeling, depending on the graph elements that are being labeled. The origin of this labeling is introduced by Rosa. The concept of labeling of graphs has gained a lot of popularity in the area of graph theory. This popularity is not only due to mathematical challenges of graph labeling but also to the wide range of applications that graph labeling offer to other branches of science, for instance, X-ray, crystallography, coding theory, cryptography, astronomy, circuit design and communication networks design. In the last three decades magic and antimagic labeling, prime labeling, graceful labeling, k-graceful labeling, and odd labeling, even and odd mean labeling and strongly labeling etc. have been studied in over 1300 papers.

PRELIMINARIES A graph is a pictorial representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges. Formally, a graph is a pair of sets (V, E), where V is the set of vertices and E is the set of edges, connecting the pairs of vertices. If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling. A Path Pn is a walk in which all the vertices are distinct. A simple graph with ‘n’ vertices (n >= 3) and ‘n’ edges is called a cycle graph if all its edges form a cycle of length ‘n’. If the degree of each vertex in the graph is two, then it is called a Cycle Graph. A wheel graph is obtained from a cycle graph Cn-1 by adding a new vertex. That new vertex is called a Hub which is connected to all the vertices of Cn. The maximum of the degrees of all the vertices is called the maximum degree of the graph and it is denoted by ∆ (G) or ∆. A simple graph in which there exists an edge between every pair of vertices is called a complete graph. A graph G is called labeled if its p points are distinguished from one another by names such as v1, v2, .….vn. A graph G is connected if for every u,v ϵ G there exists a uv-path in G. Otherwise G is called

DOI:10.21884/IJMTER.2018.5098.88NOJ 68 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 disconnected. A graph is bipartite if its vertex set is partitioned into two nonempty subsets X and Y such that each edge of G has one end in X and the other end in Y. A complete bipartite graph is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. The complete bipartite graph with partite sets of size m and n is denoted Km,n. A tree is a connected acyclic simple graph. A vertex of degree 1 is called a leaf or pendant vertex. The fan fn (n2) is obtained by joining all vertices of Pn to a further vertex called the centre and contains n+1 vertex and 2n-1 edges. A friendship graph Fn is a graph which consists of n triangle with a common vertex. A graceful labeling of a graph G of size m is an assignment of distinct elements of the set {0,1,2,…,m} to the vertices of G so that the edges labeling, which prescribes i-j to the edges joining vertices labeled i and j, assigns the labels 1,2,…,m to the edges of G. A graph possessing a graceful labeling is a graceful graph.

II. DEFINITION OF EQUALITY LABELING

Let G be a graph of order n and size m. A onto function : v (G) → {0,1,2,...,n} is called a equality labeling. If it satisfies following condition  The edges labeled by {1,2,3,…,2m-1} with │f(u)+f(v)│ if n=m.  The edges labeled by{1,2,3,…,m} with │f(u)-f(v)│if n≠m. This labeling is called an equality labeling. It is denoted by Eq (G).

III. EQUALITY LABELING FOR GENERAL GRAPHS

3.1 Path:

@IJMTER-2018, All rights Reserved 69 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 3.2 Cycle:

IV. EQUALITY LABELING FOR SPECIAL GRAPH

4.1 Diamond-Graph: A graph is diamond free if it has no diamond as an induced subgraph. The triangle free graph are diamond free graphs, since every diamond contains a triangle. 3 n≠m,n=4,m=5 1 2

3 2 1

2 1

@IJMTER-2018, All rights Reserved 70 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 4.2 Durer-Graph: Durer’s solid is combinatorially equivalent to a cube with 2 opposite vertices truncated although durer’s depiction of it is not in this from but rather as a truncated rhombohedron or triangular truncated trapezohedron. The durer graph is the graph termed by the vertices & edges of the durer solid. It is a cubic graph of girth 3 & diameter4. 4 6 2

6 3 1 4 0 10 11 1 n≠m,n= 12,m=18 2 5 3 1 8 7 1 1 2 0 3 7 10 2 4 1 6 5 9 2 7

4.3 Bull-Graph: The bull graph is a planar undirected graph with s vertices and 5 edges in the form of a triangle with 2 disjoint pendant edges. It has chromatic number 3, chromatic 3, radius3, diameter and girth 3. It is also a block graph, a split graph, an interval graph, a claw-graph, a 1-vertex connected graph and a 1- edge- connected graph. 1 2

1 7 n=m,n=5,m=5 5 0 5 3 8

3 4.4 Butterfly- Graph: A graph is bowtie-free if it has no butterfly as an induced subgraph. The triangle-free-graph are bowtie-free graph, since every butterfly contains a triangle. The full automorphism group of the butterfly graph is a group of order 8 isomorphic to the Dihedral group D4, the group of symmetries of a square including both rotation and reflection.

1 3 3 8 n≠m,n=5,m=6

8 1 0 4 2 4 2

4.5 Franklin- Graph: The Franklin graph a 3 – regular graph with 12 vertices and 18 edges. It is hamiltonion and has chromatic number 2, chromatic index 3, radius 3, diameter 3, and girth 4.It is also a 3-vertex connected and 3 edge- connected perfect graph.

2 4 2 3 4 n≠m,n=12,m=18 5 6 1 1 9 1 2 9 3 1 1 5 2 8 0 3 0 11 2 1 5 4 1 12 5

7 4.6 Herschel- Graph: The Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-hamiltonian polyhedral graph. The Herschel graph is a planar graph it can be drawn in the plane with none of its edges crossing. It is also 3-vertex connected. The removal of any 2 of its vertices leaves a connected subgraph.

4 2 8 4 2 4 6 n≠m,n=11,m=18 3 4 3 9 8 5 6 10 11 0 8 10 1 5 2 9

1 8 3 6 9

4.7 Moser- Spindle: The moser-graph (also called mosers spindle) is an undirected graph, with 7 vertices and 11 edges. It is a unit distance graph requiring 4 colors in any graph coloring.

1 4 n≠m,n=7,m=11

1 3 5 2 2 1 7 3 5 1 7 2

0 1 1

4.8 Wagner-Graph: The Wagner graph is a 3-regular graph with 8 vertices and 12edges. It is the 8-vertex mobius ladder graph.

1 4 7 5 8 n≠m,n=8,m=12 3 4

4 2 6 4 5 2 3 2 2 6 7 3 4 3 4.9 Hajos-Graph: The Hajós graph is another name for the Sierpiński sieve graph, which is isomorphic to the 2-sun graph. However, the term is also sometimes used for the

2 2 n≠m,n=6,m=9 1 4 3 1 1 4 1 4 5 3 3 0 5 5

4.10 Crown:

Crowns CnʘK1 if and only if n is even; 6

6 0 n=m,n=6,m=6

2 3

1 3 2 5 3 7 4

4.11 Chvatal-Graph: The chvatal - graph is an undirected graph with 12 vertices and 24 edges. It is triangular- free, its girth is 4 regular each vertex has exactly 4 neighbour. 12 2 10

4 7 8 5 3 7 8 n≠m,n=12,m=24 3 1 4 4 5 2 4 5 2 0 2 5 4

2 2 5 1 2

5 1 1 6 7 8 2 6

@IJMTER-2018, All rights Reserved 73 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 4.12 Goldner-Haray-Graph: The goldner is a planar graph, it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph it is also 3-vertex connected, the removal of any of its vertices leaves a connected subgraph.

5 2 2 n≠m,n=11,m=27 1 5 3 1 6

3 3 7 1 0 7 1

2 1 2 6 6 5 11 4 2 1 8 2 1 4 8 10 8 6 1 9 2

10 4.13 Errera-Graph: The Errera-Graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5-vertex –connected graph and a 5-edge connected graph. 3

4 13 n≠m,n=17,m=45 5 1 4 4 1 5 12 6 2 1 7 2 3 10 2 4 8 16 3 11 1 1 1 9 0 8 10 7 6 3 2 3 2 3 7 2 1 3 12 13 1 14 12 13 15 15 14 14 1 12 0 4 2 1

1 1 2

@IJMTER-2018, All rights Reserved 74 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 4.14 Bidiakis-Cube: The Bidiakis-cube is a cubic Hamiltonian graph and can be defined by the LCF notation [-6,4,- 4]4. The bidiakis cube can also be constructed the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. 1 3 9

2 4 9 8 0 8 6 3 1 6 5

5 5 n≠m,n=12,m=18 3 4 3 4 1 4 7 6 5 2 3 1 9 1 2 1 0 4.15 Hoffman-Graph: The Hoffman graph is a 4-regular graph with 16 vertices and 32 edges. The Hoffman graph has many common properties with the hypercube Q4 both are Hamiltonian and have chromatic number 2, chromatic index 4, girth 4and a 4-edge-connected graph. However, it is not distance regular 3

3 2 1

131 1 2 n≠m,n=16,m=32 2 0 3 1 3 1 13 1 13 12 2 14 5 16 9 3 13 16 1 7 3 0 1 0 1 0 5 4 10 15 0 0 7 3 4 3 0 3 3 3 1 3 0 3 1 3 0 4 10 7 1 3 3 3 3 0 1 0 3 2 7 0 0 10 1 0 0 1 0 9 3 2 3 1 3 0 3 6 0 0 1 3 1 0 1 0 3 1 2 3 123 1 3 6 3 6 9 1 0 3 1 1 3 1 0 2 4 11 0 3 3 3 3 8 0 1 0 0 3 1 0 1 1 1 0 4.16 Tutte-Fragment: 0 1 1 3 3 1 3 3 If this fragment part0 of 3a larger graph, then any Hamiltonian cycle through0 the graph must go in 0 0 3 3 0 or out of the top vertex. It cannot3 go1 in one1 lower vertex and out1 the other.

0 0 3 3 0 3 0 0 0 0 @IJMTER-2018, All rights0 Reserved 75

0 0 0 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 18 1 17 9 1 8 16 n≠m,n=18m=24 1

2 9 9 9 0 10 6 4 6 4 7 1 2 2 6 10 1 12 2 6 3 5 9 9 3 1 11 13 4 14 12

15 13

4.17 Sousselier-Graph: The sousselier graph is in graph theory a hypohamiltonian graph with 16 vertices and 27 edges. 15 9 6 7 8 8 4 1 1 3 4 13 12 2 14 16 15 12 6 14 16 6 n≠m,n=16,m=18 2 0 2 10 5 13 12 7 4 7 3 6 6 6 13 12 10 7 3 2 5

V. OBSERVATION

5.1 Every complete bipartite graph satisfies equality labeling.

5.2 Splitting of a star admits equality labeling.

5.3 A tree T=(V,E) of order n & maximum degree ∆ admits an equality labeling then n is even and ∆ ≤ n/2.

5.4 Let G be a complete graph, it admits equality labeling then n-∆≤ n/2.

5.5 Let G be a wheel graph with n vertices and W= {w1,w2,w3,…,wn} be a set of n weights satisfying (wm-w1)-1=m/2.

5.6 The equality labeling does not hold on loop graphs.

5.7 Every nontrivial tree is equality labeling.

@IJMTER-2018, All rights Reserved 76 International Journal of Modern Trends in Engineering and Research (IJMTER) Volume: 5, Issue: 04, [April– 2018] ISSN (Online):2349–9745 ; ISSN (Print):2393-8161 5.8 Every nontrivial path is equality labeling.

VI. NORDHAUS GADDUM RESULT

6.1 For Path Graph: For all path graphs Eq(G)+Eq(Ḡ)=Pn-1+2n-3 where n denote the number of edges in G and Pn-1 denote the edges of Ḡ.

6.2 For Cycle Graph: For all cycle graphs Eq(G)+Eq(Ḡ)=Cn-1+2(n-1) where n denote the number of edges in G and Cn-1 denote the edges of Ḡ. Also nordhaus gaddum result does not exist for C3.

6.3 For Wheel Graph: For all wheel graphs Eq(G)+Eq(Ḡ)=Wn-1+3n-5 where n denote the number of edges in G and Wn- 1 denote the edges of Ḡ.

VII. SOME RESULTS ON EQUALITY LABELING

Theorem 7.1 Pn admits Equality Labeling for all n. Proof: Let Pn be a path v1,v2,…,vn . Define a function :V(Pn){0,1,…,n} by f(vi)=i, 0≤i≤n. For every path we have n≠m. Then the edge labels are assigns Ei(1≤i≤m). It satisfies the second condition then it is labeled with │f(u)-f(v)│. Hence pn admits equality labeling. Theorem 7.2 Cn admits Equality Labeling for all n. Proof: Let Cn be a cycle v1,v2,…,vn . Define a function :V(Cn){0,1,…,n} by f(vi)=i, 0≤i≤n. For every cycle we have n═m. Then the edge labels are assigns Ei(1≤i≤2m-1). It satisfies the first condition then it is labeled with │f(u)+f(v)│. Hence Cn admits equality labeling. Theorem 7.3 Kn admits Equality Labeling for all n. Proof: Let Kn be a complete v1,v2,…,vn . Define a function :V(Kn){0,1,…,n} by f(vi)=i, 0≤i≤n. Case (i):For every complete graph we have n=m. Then the edge labels are assigns Ei(1≤i≤2m-1). It satisfies the first condition then it is labeled with │f(u)+f(v)│. In this case only K3 is satisfied and it admits equality labeling. Case (ii):For every complete graph we have n≠m. In this case Kn(n>3). Then the edge labels are assigns Ei(1≤i≤m). It satisfies the second condition then it is labeled with │f(u)-f(v)│. Hence Kn admits equality labeling. Theorem 7.4 If G admits Equality Labeling then GʘK1 admits Equality Labeling. Proof: Let G=(V, E) be a graph where V={ v i: 0 ≤ i≤ n } and E={ v i. v i+1 , 0≤ i ≤ n}. GʘK1 = (V ,E) U { v ј : 1 ≤ ј ≤ n } U { v i. v ј: 1 ≤ i , ј ≤ n }. Case (i):For every graph we have n=m. Then the edge labels are assigns Ei(1≤i≤2m-1). It satisfies the first condition then it is labeled with │f(u)+f(v)│. Hence GʘK1 admits equality labeling. Case (ii):For every graph we have n≠m. Then the edge labels are assigns Ei(1≤i≤m). It satisfies the second condition then it is labeled with │f(u)-f(v)│. Hence GʘK1 admits equality labeling. Theorem 7.5 Let G =S m,n be a double star graph then G admits admits Equality Labeling for all n.

Proof: Let G = (V, E) be a double star graph denoted by Sm,n and v1 and v2 are two vertices in Sm ,n which are not pendent. Let ui’s are m pendent vertices to vi and uj’s are n pendent vertices to v2. For every graph we have n≠m. Then the edge labels are assigns Ei(1≤i≤m). It satisfies the second condition then it is labeled with │f(u)-f(v)│. Hence Sm,n admits equality labeling.

VIII. CONCLUSION

In this paper we have investigated some general graphs and some special type of graphs which admits Equality Labeling. It is very interesting to investigate graphs which admits Equality labeling. It is possible to investigate similar results for several other graphs. Here we investigate some results corresponding to labeled graphs similar work can be carried out for other families also.

REFERENCES

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