International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017
p-ISSN: 2395-0072
www.irjet.net
MAGICNESS IN EXTENDED DUPLICATE GRAPHS K.Sutha1, K.Thirusangu2, S.Bala3 1.2,3
Department of Mathematics S. I. V. E. T. College, Gowrivakkam, Chennai – 600 073 ,Tamilnadu , India ……………………………………………………………………………………***…………………………………………………………………………….. Definition :1.2
Abstract -A graph labeling is a mapping that carries a set
of graph elements onto a set of numbers called labels(usually the set of integers). In this paper we prove the existence of graph labeling such as Z3- vertex magic total,Z3edge magic total labeling and total magic cordial labeling for extended duplicate graph of comb graph and middle graph of extended duplicate graph of path graph. We also provide an algorithm to obtain n- edge magic labeling for extended duplicate graph of comb graph .
The middle graph M(G) of a graph G(V,E) is defined with the vertex set as V E and two vertices u, v in M(G) are adjacent if they are incident in G or they are adjacent edges in G.
.Definition : 1.3
Keywords: Graph labeling, Comb graph , Path graph, Middle graph, Duplicate graph.
Let G(V,E) be a Comb graph. A Duplicate graph of G is DG=(V1,E1) where the vertex set V1 = V V' and V V'=
1 . Introduction
and f: V→V' is bijective (for v
convenience) and the edge set E1 of DG is defined as follows: The edge vivj is in E if and only if both vivj' and vi'vj are edges in E1.The extended duplicate graph of G is the graph DG { vivi′}, for some `i’.
Rosa introduced the notion of Graph labeling in 1967 [1]. In 1970 Kotzig and Rosa defined the concept of edge magic total labeling [2]. A detailed study on graph labeling has been done by Gallian [3 ]. MacDougall et. al introduced the notion of vertex magic total labeling in 1999 [4].
Definition : 1.4 A labeling function f : V E → Z3 -{0} is said to be a Z3
Thirusangu et., al introduced the concept of Extended Duplicate graph [5]. They proved that the Extended duplicate graph of twig graph admits Z3 – vertex magic total , edge magic total and total magic cordial labeling. In [6] they also proved some results on Extended duplicate graph of comb graph.
– vertex magic total labeling in G(V,E) if there exist a function f*: V → N {0} such that f*(vi) = {f(vi) +∑ f(e)} (mod 3) , is a constant where e is the edge incident at vi .
Definition : 1.5
In 2012 Jeyapriya and Thirusangu introduced 0- Edge magic labeling and shown the existence of this labeling for some class of graphs [7].Neelam Kumari and Seema Mehra establish the concept of 1- Edge magic and n – Edge magic labeling [8].They proved Pt ,Ct(t is even) and sun graph St(t is even) are n- Edge magic.
A labeling function f : V E → Z3 -{0} is said to be a Z3– edge magic total labeling in G(V,E) if there exist a function f*: E → N {0} such that f*(vivj) ={f(vi+f(vj)+f(vivj)}(mod3) is a constant for all edges vivj E .
Hamada et. ,al introduced Middle graph [9] and they proved middle graph of the complete graph ( Kn) has (n-1) forest coloring. Arundhadi and Thirusangu proved some colorings on middle graph of some class of graphs [10].
Definition : 1.6 A graph G(V,E) is said to admit total magic cordial labeling if f : V E → {0,1} such that (i). f(x) + f(y)+f(xy)}
Definition: 1.1
(mod 2) is constant for all edges xy
Let Pm+1 be a path graph .Comb graph is defined as Pm+1ʘ (m+1)K1. It has 2m+2 vertices and 2m+1 edges.
© 2017, IRJET
|
Impact Factor value: 5.181
V, we write f(v) = v' for
E. (ii) for all i,j ϵ
{0,1} , { mi(f) + ni(f) } – { mj(f) + nj(f) } ≤ 1, (i ≠ j) where mi(f) = {e E / f(e) = i} and ni(f) = {v V/ f(v) = i}
|
ISO 9001:2008 Certified Journal
|
Page 364