CHAPTER I
PRELIMINARIES
Introduction: The structure of graphs admits a variety of functions that
assign real number to their vertices and edges so that certain given
conditions are satisfied. Numbered undirected graphs are becoming an
increasingly useful family of mathematical models for a broad range of
applications.
If a non negative integer f(e) is assigned to each edge e, then the edges
of G are said to be numbered. Then each vertex v is given value
f + (v) ≡ ( Σf (uv) ) (mod (2k)), where this sum run over all edges through v.
Thus every graph can be numbered in infinitely many ways if the additional
conditions are absent. Thus the numbered graph models are used after
imposing additional conditions.
Rosa [1967] introduced graph labeling, later on called as graceful
labeling. The graceful labeling problem is to determine which graphs are
graceful. Given a graph G consisting of vertices and edges, a vertex labeling
of G is an assignment f of labels to the vertices of G that produces for each
edge xy a label depending on the vertex labels f(x) and f(y).
A vertex labeling f is called a graceful labeling of a graph G with e
edges if f is an injection from the vertices of G to the set {0, 1 , . . . , e} such
that when each edge xy is assigned the label │f(x) - f(y)│, the resulting edge
labels are distinct. A graph G is called graceful if there exists a graceful
labeling of G.
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Rosa [1967] has identified three reasons why a graph fails to be
graceful: (1) G has “too many vertices” and “not enough edges,” (2) G “has
too many edges,” and (3) G “has the wrong parity.” An
finite
in class of
graphs that are not graceful for the second reason is given in [1986]. As an
example of the third condition, Rosa has shown that if every vertex has even
degree and the number of edges is congruent to 1 or 2 (mod 4), then the
graph is not graceful. In particular, the cycles C 4n+1 and C 4n+2 are not
graceful.
Golomb [1972] further studied such labeling. Sheppard [1976] has
verified that there are exactly q! gracefully labeled graphs with q edges.
A complete summary of graceful, non-graceful graphs and the results along
with some unproven conjectures can be found in Gallian’s Dynamic Survey
[2010] of graph labeling. The survey reveals that the gracefulness of several
classes of graphs has already been established. For example, all the paths P n ,
all the trees, the complete graphs K n for n < 5, all the wheels, etc., are
graceful. While the graceful labeling of graphs is perceived to be a primarily
theoretical subject in the field of graph theory and Discrete Mathematics,
gracefully labeled graphs often serve as models in a wide range of
applications.
Such applications include coding theory, communication network,
X-ray Crystallography, Radar, Astronomy, Circuit Design, Data Base
Management, and models for constraint programming over finite domains.
Bloom and Golomb [1977] gave a detailed explanation of some of the
applications of gracefully labeled graphs.
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This chapter contains some basic definitions, preliminaries and a
brief summary of results obtained on the gracefulness, edge – odd
gracefulness and edge-vertex-even gracefulness of various graphs.
1.1 Section – Preliminaries and Previous concepts:
In this section, various types of graceful labeling are presented.
A number of authors have invented analogues of graceful graphs by
modifying the permissible vertex labels as well as edge labels. For instance,
Lee [1991] defined Skolem - graceful labeling of a graph G with p vertices
and q edges if there is an injection from the set of vertices of G to
{1, 2, …, p} such that the edge labels induced by f(x) - f(y) for each edge
xy are {1, 2, …, q}. A necessary condition for a graph to be Skolem-graceful
is that p ≥ q + 1.
Lo [1985] introduced similar kind of labeling called edge-graceful. A
graph G(V, E) is said to be edge-graceful if there exists a bijection f from E
to {1, 2, …,E} such that the induced mapping f+ from V to{0,1,…,V-1}
given by f+(x) ≡ (∑f(xy)) (mod V) taken over all edges xy is a bijection.
Gnanajothi [1991] has defined line-graceful similar to edge-graceful.
A graph with n vertices is called line-graceful if it is possible to label its
edges with 0, 1, 2, … , n such that when each vertex is assigned the sum
modulo n of all the edge labels incident with that vertex, the resulting vertex
labels are 0, 1, 2, … , (n – 1). A necessary condition for the line-gracefulness
of a graph is that its order is not congruent to 2 (mod 4).
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Also Gnanajothi [1991]defined a graph G with q edges to be
odd-graceful if there is an injection f from V (G) to {0, 1, 2, …, 2q – 1} such
that, when each edge xy is assigned the label f(x) - f(y), the resulting edge
labels are {1, 3, 5, …, 2q – 1}.
Van Bussel [2002] considered two kinds of relaxations of graceful
labelings. He called a labeling range-relaxed graceful. It meets the
conditions same as that of a graceful labeling except the range of possible
vertex labels. Also edge labels are not restricted to the number of edges of
the graph (the edges are distinctly labeled but not necessarily labeled 1 to q
where q is the number of edges). Similarly, he called a labeling as
vertex-relaxed graceful if it satisfies the conditions of a graceful labeling
while permitting repeated vertex labels.
Sekar [2002] called an injective function f from the vertices of a graph
with q edges to {0, 1, 3, 4, 6, 7, …, 3(q - 1), (3q – 2)} one modulo three
graceful if the edge labels induced by labeling each edge uv with
f(u) - f(v) is {1, 4, 7, …, 3q – 2}.
Lee, Pan, and Tsai [2005] called a graph G, with p vertices and
q edges, vertex-graceful if there exists a labeling f: V(G) → {1, 2, …, p}
such that the induced labeling f+ from E(G) to 2q defined by
f+(uv) = [f(u) + f(v)] (mod q) is a bijection. Vertex-graceful graphs can be
viewed as the dual of edge-graceful graphs.
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Another labeling called even-edge graceful was introduced in [2007].
A (p, q)-graph with q ≥ p is an even edge graceful if there is an injection f
from the set of edges to {1, 2, 3, …, 2q} such that the values of the induced
mapping f+ from the vertex set to {0, 1, 2, …, 2q–1} given by
f+ (x) = (∑f(xy)) (mod 2q) over all edges xy are distinct and even. The same
is generalized as even-vertex graceful by A.Solairaju and P. Muruganandam.
A.Solairaju and K.Chitra [2008] introduced the concept of edge-odd
graceful labeling. A (p, q) connected graph is edge-odd graceful graph if
there exists an injective map f: E(G) → {1, 3, …, 2q-1} so that induced map
f + : V(G)→{0, 1,2, 3, …,(2k -1)} defined by f + (x) ≡ (∑f(x, y)) (mod 2k),
where the vertex ‘x’ is incident with other vertex y and k = max {p, q},
makes the resulting labels distinct and odd.
Some definitions that are needed are furnished below.
1.1.1 Definition: A graph is a set of object called nodes or vertices or points
connected by links called lines or edges. The order of a graph is the number
of vertices denoted by |V| or p. A graph size is the number of edges denoted
by |E| or q. A graph with p vertices and q edges is called a (p, q) graph.
The degree of a vertex is the number of edges incident through it. If two
distinct edges e 1 and e 2 are incident with a common vertex, then they are
adjacent edges.
1.1.2 Definition: A graph is said to be finite if the order p (or q) is finite.
A graph is called simple if there is neither loop (an edge that has the same
end points) nor multiple edges (more than one edge between two vertices).
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Every graph mentioned in this thesis is a simple finite graph.
1.1.3 Definition: The edges of a graph represent connections directly from
one vertex or node to another. A walk is a sequence v 1 e 1 v 2 e 2 ,…..e k-1 v k
alternating between vertices v i and edges e i , such that each edge e i is
incident to the two vertices v i and v i+1 . A walk such that all the edges e i are
distinct is called a trail.
1.1.4 Definition:
A path P n of length (n -1) is a subgraph of graph
G = (V, E) such that a sequence of distinct edges v 1 v 2 ,v 2 v 3 ,…,v n-1 v n with
v i ∈ V(G) and v i v i+1 ∈ E(G) for each i = 1,2,….,n. P n is called a closed path
if v I = v n . It is also called a cycle or a circuit, and it is denoted by C n .
1.1.5 Definition: A connected graph G = (V, E) is a graph in which there is
a path from x to y for every pair of vertices x, y∈V. Otherwise, it is called
disconnected.
1.1.6 Definition: Two graphs G 1 and G 2 are called vertex disjoint graphs if
V(G 1 ) ∩ V(G 2 ) = { }. Let G 1 and G 2 be two vertex disjoint graphs. A union
of G 1 and G 2 denoted by G = G 1 ∪ G 2 , is the graph that consists of
V (G) = V(G 1 ) ∪ V(G 2 ) and E(G) = E(G 1 ) ∪ E(G 2 ).
1.1.7 Definition: The Cartesian product of two graphs G and H, written as
G
H, is the graph with vertex set V(G) × V(H) specified by putting (g, h)
adjacent to (g′, h′) if and only if (1) g = g′ and hh′∈ E(H) , or (2) h = h′ and
gg′∈ E(G).
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1.1.8 Definition: A spanning tree of a connected, undirected graph is a tree
that includes every vertex of that graph.
1.1.9 Definition: The graph n o C 3 is a connected graph which has one edge
union of n number of C 3 .
1.1.10 Definition: The friendship graph F n is a connected graph obtained by
taking n copies of the cycle graph C 3 with a vertex in common.
1.1.11 Definition: The Dutch windmill graph D m (n) is the graph obtained by
taking n copies of the cycle graph C m with a vertex in common.
1.1.12 Definition: The graph P m + N n is a connected graph such that every
vertex of P m is adjacent to every vertex of null graph N n together with
adjacent edges in P m .
1.2 Section: Cycle related graphs
Here the edge – odd gracefulnesses of graphs related to cycle with
path and star are obtained.
Introduction: Rosa [1967] proved that the n-cycle C n is graceful if and only
if n ≡ 0 or 3 (mod 4). Liu [1996] has shown that if two or more vertices are
inserted between every pair of vertices of the n-cycle of the wheel W n , the
resulting graph is graceful. Sekar [2002] verified that the graph obtained by
identifying an endpoint of a star with a vertex of a cycle is graceful.
Eldergill [1997] got that the one point union of any number of copies of C 6
is odd-graceful.
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In 2008 – 2010, A.Solairaju, and K. Chithra found edge – odd
graceful labeling for the following graphs: The path P n for n ≥ 3; C n for odd
n ≥ 3; C n Θ P m for n, m ≥ 3; B(C n , C n ), n ≥ 3; crown graph C n K 1 for n ≥
3; crown graph with one edge deleted C n K 1 \{e} for n ≥ 3; one edge added
〈C n K 1 \{e}〉 for n ≥ 3 and crown with parallel chords CP(n) for n ≥ 4;
the cycle C n with parallel chords n ≥ 4; C n (m) where both n and m are odd or
both n and m are even; the product graph P n x C 3 for n ≥ 2.
1.2.1 Definition: C m Θ S n+1 is a connected graph obtained from the circuit
C m by adding a star graph S n+1 to each of the m vertices of C m . It has
(mn+m) vertices and (mn+m) edges.
1.2.2 Definition: Armed crown C m Θ P n is a connected graph obtained from
the circuit C m by adding a path P n to each vertex of C m . It has mn vertices
and mn edges.
1.2.3 Definition: Bi-armed crown C m Θ 2P n is a connected graph obtained
from the circuit C m by adding two paths P n to each vertex of C m . It has
m(2n-1) vertices and m(2n-1) edges.
1.2.4: In chapter II, the following graphs are proved as edge – odd
graceful.
1: The connected graph C m Θ S n where n ≥ 2 is even; all positive integer m.
2: The connected graph C 3 Θ P n for n ≥ 2.
3: The connected graph C 3 Θ 2P n for n ≥ 2.
4: The connected graph C 3 Θ 3P n .
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5: The connected graph C 5 Θ P n for n ≥ 2.
6: The connected graph C 5 Θ 2P n for n ≥ 2.
7: The connected graph C 7 Θ P n for n ≥ 5.
8: The connected graph C m Θ 2P n for odd m and n ≥ 1.
1.2.5 Example: The following graphs are given as examples for edge-odd
graceful graphs due to various rules in the above graphs.
1.C 3 Θ S 6 ;
2. C 3 Θ P 2 ;
3. C 3 Θ P 3 ;
4.C 3 Θ P 4 ;
5.C 3 Θ P 5 ;
6. C 3 Θ P 6 ;
7. C 3 Θ 2P 4 ;
8. C 3 Θ 3P 3 ;
9. C 5 Θ P 5 ;
10. C 5 Θ P 6 ;
11. C 5 Θ 2P 4 ;
12. C 5 Θ 2P 5 ;
13. C 7 Θ P 6 ;
14. C 7 Θ 2P 4 .
1.3 Section: Star related graphs
In this section, edge–odd gracefulness of the graphs related to star is found.
Introduction: Sekar [2002] proved that the graph obtained by identifying an
endpoint of a star with a vertex of a cycle is graceful. The book B m is the
graph S m ∪ P 2 where S m is the star with (m + 1) vertices. Maheo [1980]
showed that the books of the form B 2m are graceful and conjectured that the
books B 4m+1 were also graceful.
Wu [2002] verified that if G is a graceful graph with n edges and
(n+1) vertices then the join of G and any star are graceful. Sethuraman and
Selvaraju [2001] established that every supersubdivision of a path is
graceful. They conjectured that every supersubdivision of a star is graceful.
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They also got that paths and stars are the only graphs for which every
supersubdivision is graceful. The conjecture that every supersubdivision of a
star is graceful was proved by Kathiresan and Amutha in [2004].
A. Solairaju and K.Chitra in [2008 – 2010] has found edge – odd
graceful labeling for the following star related graphs: S n ∪ S m where n and
m are even; S n ∪ S m ∪ S r where n, m and r are even; P 3 ∪ S n , P 4 ∪ S n and
C 4 ∪ S n for even n; bi-stars B n,n for odd n; 〈K 1, n : 2〉 for odd n; double star
K 1, n, n for n ≥ 2; triple star K 1, n, n, n for
n ≥ 3; caterpillar C(n, n-1, m) for
odd n and for m ≥ 2; n*C 4 ; n◦C 4 ; C(n, n-3)◦S n ; C(n, n-3) □ S n ; P 2 (+) P n and
P 3 (+)P n ; P 2 (+) N 2n and P 3 (+)N 2n .
1.3.1 Definition: The book graph S 2 □S n (or Cartesian product of the star
graphs S 2 and S n ) is a connected graph obtained by adding ‘n’ number of C 4
with one edge. It has (2n) vertices and (3n – 2) edges.
1.3.2 Definition: Double star graph S 2, n is a tree obtained from the star S 1, n
by adding a new pendent edge to each of the existing n pendent vertices. It
has (2n+1) vertices and 2n edges.
1.3.3 Definition: The m star graph S m, n is a tree obtained from the double
star graph S 2, n by merging a path P (m-2) to each of the existing n pendent
vertices. It has (mn+1) vertices and mn edges.
1.3.4 Definition: The graph JE m,n is defined as having a center node, n legs
of length 2 and m legs of length 1. It consists of (2n + m + 1) vertices and
(2n + m) edges.
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1.3.5: The concept of edge-odd graceful labeling is extended for the
product graph, and the results are given below.
9: The connected graph S 2 □S n , for n ≥ 3, is edge – odd graceful.
10: The connected graph S 2, n is edge – odd graceful for n > 1.
11: The connected graph S m, n is edge – odd graceful for all m, n >1.
12: The connected graph JE 3, n is edge – odd graceful for n >1.
1.3.6: Examples of edge odd graceful graphs are given for
1. S 2 □S 7 for n is odd;
2. S 2 □S 6 for n ≡ 0 (mod 6);
3. S 2 □S 8 for n ≡ 2 (mod 6);
4. S 2 □S 10 for n ≡ 4 (mod 6);
5. S 2, 8 ;
6. S 2, 9 ;
7. S 6, 5, m ≡ 2 (mod 4) and n ≡ 1 (mod 4);
8. S 6, 7, m ≡ 2 (mod 4) and n ≡ 3 (mod 4);
9. S 8, 5, m ≡ 0 (mod 4) and n ≡ 1 (mod 4);
10. S 8, 7, m ≡ 0 (mod 4) and n ≡ 3(mod 4);
11.S 7, 5, m ≡ 3 (mod 4) and n ≡ 1 (mod 4);
12. S 7, 7, m ≡ 3 (mod 4) and n ≡ 3 (mod 4);
13. S 9, 5, m ≡ 1 (mod 4) and n ≡ 1 (mod 4);
14. S 9, 7, m ≡ 1 (mod 4) and n ≡ 3 (mod 4);
15. S 6, 8, m ≡ 2 (mod 4) and n ≡ 0 (mod 4);
16. S 4, 8, m ≡ 0 (mod 4) and n ≡ 0 (mod 4);
17. S 7, 8, m ≡ 3 (mod 4) and n ≡ 0(mod 4);
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18.S 5, 6, m ≡ 1 (mod 4) and n ≡ 2 (mod 4);
19. S 6, 6, m ≡ 2 (mod 4) and n ≡ 2 (mod 4);
20. JE 3,4, n ≡ 0 (mod 4);
21. JE 3,5, n ≡ 1 (mod 4);
22. JE 3,6, n ≡ 2 (mod 4);
23. JE 3,7, n ≡ 3 (mod 4).
1.4 Section - Graphs with the combination of Path and Star:
Here, the concept of edge-odd graceful is extended to the graphs with
the combination of path and star.
Introduction: Chen, Lu, and Yeh [1997] proved that firecrackers are
graceful and conjectured that banana trees are graceful. Sethuraman and
Jesintha [2009] and [2005] have shown that all banana trees and extended
banana trees (graphs obtained by joining a vertex to one leaf of each of any
number of stars by a path of length of at least two) are graceful. Gao [2007]
has verified that the union of any number of stars and paths are odd graceful.
A.Solairaju and K.Chithra [2008 – 2010] found edge – odd graceful
labeling for the following path related graphs:
The path P n for n ≥ 3; sunflower graph SF(n) for n ≥ 3; fan graph F n
for n ≥ 4; double fan Df n = P n + K 2 for 2 ≤ n ≤ 9; the product graph P n x C 3
for n ≥ 2; the ladder graph L n = P n x P 2 for n ≥ 3; the prism D n for n ≥ 3;
P n  P 2 for n ≥ 2; P n ∪ P m where n and m are odd. P 2 + P n for n = 3, 4, 6,
10, 14 … ; P 3 + P n where n is even; P 2 + N n for even n; P 3 (+) N n where n is
even.
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1.4.1 Definition: The fire - cracker graph P m Θ S 5 is a tree obtained from
the path P m by adding a star graph S 5 to each of the m vertices of P m . It has
(6m) vertices and (6m - 1) edges.
1.4.2 Definition: The graph P m Ο S n+1 is a tree obtained from the path P m by
adding a star graph S n to each of the pendent vertices of P m . It has
(2n + m) vertices and ((2n + m - 1) edges.
1.4.3 Definition: P m + P n is a connected graph such that every vertex of P m
is adjacent to every vertex of graph P n together with adjacency in both
P m and P n . It has (n+m) vertices and [(m+1)n + (m-2)] edges.
1.4.4: The edge-odd graceful labeling is verified for the graph with path
and star, and the results are furnished below:
13: The connected graph P m Θ S 5 is edge – odd graceful for n > 1.
14: The connected graph P 5 Ο S n is edge – odd graceful for n > 1.
15: The connected graph P 4 Ο S (2n-1) is edge – odd graceful for n > 1.
16: The connected graph P 6 Ο S n is edge – odd graceful for n > 1.
17: The connected graph P 7 Ο S n is edge – odd graceful for n > 1.
18: The connected graph P 8 Ο S n is edge – odd graceful for odd n > 5.
1.4.5: Examples of edge odd graceful graphs are given for path with star
1. P 4 Θ S 5 ; P 7 Θ S 5 ;
2. P 4 Ο S 7 ;
3. P 5 Ο S 9 ; P 5 Ο S 11 ;
4. P 6 Ο S 9 ;
5. P 7 Ο S 5, for n ≡ 3, 5 (mod 8);
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6. P 7 Ο S 7, for n ≡ 1, 7 (mod 8);
7. P 8 Ο S 3 ,for n ≡ 0 (mod 8);
8. P 8 Ο S 5, for n ≡ 5 (mod 8).
1.4.6: The edge-odd graceful labeling is verified for the join of paths,
and the results are furnished below:
19: The connected graph P 2 + P n is edge – odd graceful.
20: The connected graph P 3 + P n for n = 1, 2, …, (4n + 1) is edge – odd
graceful.
21: The connected graph P 4 + P n for n = 1, 2, 3, 4, …, (5n + 2) is edge – odd
graceful.
22: The connected graph P 5 + P n for all n and n ≠ 7 is edge – odd graceful.
23: The connected graph P 6 + P n for all n and n ≠ 7 and 8 is edge – odd
graceful.
24: The connected graph P n + P n for all n > 1 is edge – odd graceful.
1.4.7: Examples of edge-odd graceful graphs are given for join of paths
1. P 2 + P 6 ;
2. P 3 + P 3 ; P 3 + P 4 ; P 3 + P 5 ;
3. P 4 + P 4 ; P 4 + P 5 ;
4. P 5 + P 7 ;
5. P 6 + P 7 ; P 6 + P 8 ;
6. P 5 + P 5 ; P 6 + P 6 ;
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1.5 Section - Shell graph C(n, n-3) and Clock graph (2C n + nP 2 ):
In this section, the edge-odd graceful labeling for Shell graph and
Clock graph (2C n + nP 2 ) is obtained.
Introduction: In [2008], A. Solairaju and K. Chitra proved that the shell
graph C(n, n-3) merging with a star graph is graceful. In 2008 – 2010, they
also found edge – odd graceful labeling for the following graphs:
B(C n , C n ), n ≥ 3; the cycle C n with parallel chords n ≥ 4; the product graph
P n x C 3 for n ≥ 2.
1.5.1 Definition: A shell graph of size n, denoted by C(n, n-3), is the graph
obtained from the cycle C n (v 1 , v 2 , …, v n ) by adding (n-3) consecutive chords
incident with a common vertex. Such a graph G is called one edge union of
k (≥ 1) shell graphs C(n, n-3).
1.5.2: Further result on edge-odd graceful labeling is verified for the
shell graphs 2C(n, n-3) and 3C(n, n-3) and is furnished here.
25: One edge union of shell graph 2C(n, n-3) is edge odd graceful.
26: One edge union of shell graph 3C(n, n-3) is edge odd graceful.
1.5.3: Examples of edge odd graceful graphs are given for one edge union
of shell graph
1. 2C(7, 4);
2. 3C(7, 4).
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1.5.4 Definition: Let 2C n be 2 copies of circuits of length n. Let the first
copy of C n has vertices as v 1 , v 2 , …, v n and second copy of C n has vertices
w 1 , w 2 , …, w n . The ring 2C n + nP 2 (Clock graph ) is 2 C n graph
as
such that
v 1 w 1 , v 2 w 2 , …, v n w n are all adjacent. The graph 2C n is
disconnected but
2C n + nP 2 is connected graph. It has 2n vertices and 3n
edges.
1.5.5: The edge-odd graceful labeling is verified for the Clock graph
27: The connected graph 2C n + nP 2 is edge – odd graceful.
1.5.6: Examples of edge odd graceful graphs are given for Clock graph
2C n + nP 2
1. 2C 7 + 7P 2 , n ≡ 2 (mod 3);
2. 2C 8 + 8P 2 , n ≡ 1 (mod 3);
3. 2C 6 + 6P 2 , n ≡ 0 (mod 3).
1.6 Section - Edge–Vertex–Even Gracefulness of some graphs:
Also, in this section, a new method of labeling called edge–vertex–
even graceful labeling is introduced and is verified for some graphs.
Introduction: After the introduction of graceful labeling for the graphs by
Rosa in [1967], several authors invented various labeling by imposing
different conditions to the edges as well as to the vertices.
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Lo [1985] introduced similar kind of labeling called edge-graceful.
Gnanajothi [1991] developed a concept similar to edge-graceful called linegraceful. Also in 1991, Gnanajothi defined another labeling called oddgraceful. Lee, Pan, and Tsai [2005] found another graph labeling called
vertex-graceful. Vertex-graceful graphs can be viewed as the dual of edgegraceful graphs. The concept of even- edge graceful was found in [2007].
A.Solairaju and K.Chitra in 2008 introduced the concept of edge-odd
graceful labeling. These different labelings motivate us to study another kind
of labeling called edge-vertex-even graceful.
Edge-vertex-even gracefulness of the graph is defined as follows:
A (p, q) connected graph is edge-vertex-even graceful graph if there
exists an injective map f: E(G) → {2, 4, …, 2q} so that induced map
f + : V(G) → {0, 2, 4, …, (2k-2)} defined by f + (x) ≡ (Σf(x, y))(mod 2k),
where the vertex x is incident with other vertex y and k = max {p, q}, makes
the resulting labels distinct and even. When the graph admits the labeling of
edge-vertex-even graceful, the graph is called edge-vertex-even graceful
graph.
1.6.1 Graceful labeling is verified for the following graph.
28: The spanning tree of the graph of Cartesian product of S m and S n .
1.6.2 The edge-vertex-even graceful labeling is obtained for some graphs
and the results are furnished below:
29: The path P n is edge–vertex–even graceful for odd n.
30: The circuit graph C n is an edge–vertex–even graceful for odd n.
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31: The star graph S n ( or K 1, n-1 ) is an edge–vertex–even graceful for odd n.
32: The n o C 3 is edge–vertex–even graceful where n is even.
33: The graph P 2 □ C n is edge–vertex–even graceful.
34: A spanning tree of the Cartesian product graph S 3 □ S n is edge–vertex–
even graceful.
35: The connected graph 2C n □ nP 2 is edge–vertex–even graceful.
36: The connected graph S n+1 + P n is edge–vertex–even graceful.
37: The connected graph P 2 + P n is edge–vertex–even graceful.
38: The connected graph P 2 + N n is edge–vertex–even graceful.
39: The connected graph P 3 □ Nn is edge–vertex–even graceful.
40: The friendship graph F n is edge–vertex–even graceful.
41: The connected graph D 4 n for odd n is edge–vertex–even graceful.
1.6.3: Examples of edge-vertex-even graceful graphs are given below:
1. A spanning tree of the Cartesian product of S 2 and S 6 ;
2. A spanning tree of the Cartesian product graph S 3 □ S 3 ;
3. The path P 9 ;
4. The circuit C 9 ;
5. The star S 7 (or K 1, 6 );
6. 6 o C 3 ;
7.P 2 □ C 4 ;
8.The connected graph 2C 7 + 7P 2 and 2C 8 + 8P 2 ;
9. The connected graph S 8 + P 7 ;
10. The connected graph S 10 + P 9 ;
11.The connected graph P 2 + P 10 for n≡ 4(mod 6);
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12. The connected graph P 2 + P 6 for n≡ 2(mod 6);
13. The connected graph P 2 + P 5 for n≡ 1, 3, 5(mod 6);
14. The connected graph P 2 + P 6 for n≡ 0(mod 6);
15. The connected graph P 2 + N 8 for n ≡ 0, 3, 5, 6 (mod 8);
16. The connected graph P 2 + N 10 for n≡ 1, 2, 4, 7 (mod 8);
17. The connected graph P 3 + N 10 ;
18. Friendship graph F 6 for n ≡ 0 (mod 3);
19. Friendship graph F 7 for n ≡ 1, 2 (mod 3);
20. Wind mill graph D 4 5 and D 4 7.
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