Volume 4, Issue 6, June – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 IJISRT19JU676 www.ijisrt.com 795 Cycle and Path Related Graphs on L – Cordial Labeling 1 J. Arthy Department of Mathematics 1 Ph.D Scholar, Mother Teresa Women’s University, Kodaikanal, Tamilnadu, India 2 Manimekalai K BIHER, Chennai, Tamilnadu, India. 3 Ramanathan. K KCG College of Engineering and Technology, Chennai, Tamilnadu, India Abstract :- In this work we establish TLn , DLn , Qn , DQn , ITn ,K1+K1,n, The generalized antiprism m n A , Pn ʘK1, Hn , Hnʘ , , Duplication of all edges of the Hn , Braid graph B(n) , Z- Pn , The duplication of every edge by a vertex in Cn are L – Cordial . Keyword:- L – Cordial Labeling (LCL), L – Cordial Graphs (LCG), Ladder, Snake, Path and Corona Graphs. AMS Classifications: 05C78 I. INTRODUCTION L – Cordial Labeling (LCL) was introduced in [7]. In [8,9] they discussed LCL behaviour of some standard graphs. Prime cordial, Cube difference and Square difference labeling of H- related graphs has been studied in [1,4,12]. 4 – Cordiality of path related graphs is investigated in [10]. Pairsum labeling of star and cycle related graphs, Prime labeling of duplication graphs, Difference Cordiality of ladder and snake related graphs, Super Mean labeling of antiprism and some more graphs have been proved in [5,6,11,13]. For this study we use the graph G= (p.q) which are finite, simple and undirected. A detailed survey of graph labeling is given in [3]. Terms and results follow from [2]. In this work we study some standard and special graphs are LCG. Definition 1.1[7] Graph G (V,E) has L-cordial labeling if there is a bijection function f :E(G) E ,... 2 , 1 .Thus the vertex label is induced as 0 if the biggest label on the incident edges is even and is induced as 1, if it is odd. The condition is satisfied further by ) 0 ( f V which number of vertices labeled with 0 and ) 1 ( f V which is the number of vertices labeled with 1, and follows the condition that . 1 ) 1 ( ) 1 ( f f V V Isolated vertices are not included for labeling here. A L- cordial graph is a graph which admits the above labeling. Definition 1.2[11] The triangular ladder (TLn) is obtain from a ladder by including the edges 1 i i u v for i =1, 2...n-1 with 2n vertices and 4n-3 edges. Definition 1.3[11] A diagonal ladder (DLn) is a graph formed by adding the vertex of i v with 1 i u and i u with 1 i v for 1 1 n i . Definition 1.4[9] Qn is said to be quadrilateral snake if each edge of the path Pn is replaced by a cycle C4. Definition 1.5[9] A double quadrilateral snake DQn consist of two quadrilateral snake that have a common path. Definition 1.6[9] The irregular triangular snake ITn is derived from the path by replacing the alternate pair of vertices with C3. Definition 1.7[5] m n A is the generalized antiprism formed by generalized prism Cn×Pm by adding the edges 1 j i j i v v for n i 1 and 1 1 m j . Definition 1.8[1] Hn -graph obtained from two copies of path with vertices 1 , 2 ,…, and 1 , 2 ,…, by connecting the vertices 2 1 n a and 2 1 n b if n is odd and 1 2 n b and 2 n a is joined if n is even. Definition 1.9[3] The corona G1ʘG2 is defined as the graph G obtained by taking one copies of G1(which has p points) p copies of G2 and then joining the i th point of G1 to every point in the i th copy of G2. Definition 1.10[13] Duplication of an edge e = xy of a graph G produces a new graph G ’ by adding an edge e’ = x ’ y ’ such that N( x ’ ) = N(x) ∪( y ’ ) –{y} and N(y ’ ) = N(y) ∪ ( x ’ ) – {x}.
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Cycle and Path Related Graphs on L Cordial Labeling
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Volume 4, Issue 6, June – 2019 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
IJISRT19JU676 www.ijisrt.com 795
Cycle and Path Related Graphs on
L – Cordial Labeling
1J. Arthy
Department of Mathematics 1Ph.D Scholar, Mother Teresa Women’s University,
Kodaikanal, Tamilnadu, India
2Manimekalai K
BIHER, Chennai, Tamilnadu,
India.
3Ramanathan. K
KCG College of Engineering and Technology,
Chennai, Tamilnadu, India
Abstract :- In this work we establish TLn , DLn , Qn ,
DQn , ITn ,K1+K1,n, The generalized antiprismm
nA , Pn
ʘK1, Hn , Hnʘ 𝑲𝟏,𝒎̅̅ ̅̅ ̅̅ , Duplication of all edges of the Hn ,
Braid graph B(n) , Z- Pn , The duplication of every edge
by a vertex in Cn are L – Cordial .
Keyword:- L – Cordial Labeling (LCL), L – Cordial
Graphs (LCG), Ladder, Snake, Path and Corona Graphs.
AMS Classifications: 05C78
I. INTRODUCTION
L – Cordial Labeling (LCL) was introduced in [7]. In
[8,9] they discussed LCL behaviour of some standard
graphs. Prime cordial, Cube difference and Square
difference labeling of H- related graphs has been studied in
[1,4,12]. 4 – Cordiality of path related graphs is
investigated in [10]. Pairsum labeling of star and cycle
related graphs, Prime labeling of duplication graphs,
Difference Cordiality of ladder and snake related graphs,
Super Mean labeling of antiprism and some more graphs
have been proved in [5,6,11,13]. For this study we use the
graph G= (p.q) which are finite, simple and undirected. A
detailed survey of graph labeling is given in [3]. Terms and
results follow from [2]. In this work we study some
standard and special graphs are LCG.
Definition 1.1[7]
Graph G (V,E) has L-cordial labeling if there is a
bijection function f :E(G) E,...2,1 .Thus the vertex label
is induced as 0 if the biggest label on the incident edges is
even and is induced as 1, if it is odd. The condition is
satisfied further by )0(fV which number of vertices labeled
with 0 and )1(fV which is the number of vertices labeled
with 1, and follows the condition that .1)1()1( ff VV
Isolated vertices are not included for labeling here. A L-
cordial graph is a graph which admits the above labeling.
Definition 1.2[11]
The triangular ladder (TLn) is obtain from a ladder by
including the edges 1iiuv for i =1, 2...n-1 with 2n
vertices and 4n-3 edges.
Definition 1.3[11]
A diagonal ladder (DLn) is a graph formed by adding
the vertex of iv with 1iu and iu with 1iv for
11 ni .
Definition 1.4[9]
Qn is said to be quadrilateral snake if each edge of the
path Pn is replaced by a cycle C4.
Definition 1.5[9]
A double quadrilateral snake DQn consist of two
quadrilateral snake that have a common path.
Definition 1.6[9]
The irregular triangular snake ITn is derived from the
path by replacing the alternate pair of vertices with C3.
Definition 1.7[5] m
nA is the generalized antiprism formed by
generalized prism Cn×Pm by adding the edges 1j
i
j
i vv for
ni 1 and 11 mj .
Definition 1.8[1]
Hn -graph obtained from two copies of path with
vertices 𝑎1, 𝑎2, … , 𝑎𝑛 and 𝑏1, 𝑏2, … , 𝑏𝑛 by connecting the
vertices
2
1na and
2
1nb if n is odd and 1
2
nb and
2
na is joined
if n is even.
Definition 1.9[3]
The corona G1ʘG2 is defined as the graph G obtained
by taking one copies of G1(which has p points) p copies of
G2 and then joining the ith point of G1 to every point in the
ith copy of G2.
Definition 1.10[13]
Duplication of an edge e = xy of a graph G produces a
new graph G’ by adding an edge e’ = x’y’ such that