CHAPTER 11
PRELIMINARIES
In this chapter we collect the basic definitions
and theorems on graphs .and digraphs which are
needed for the subsequent chapters.
Definition 1.1 A. graph G = (V, E) is a finite non-empty set V of objects
called vertices together with a set E of unordered pairs of distinct vertices called
edges.
The edge e = uv is said to join the vertices u and v and we say
that u and v are adjacent vertices, u and e are incident as are v and e. If e1 and
e2 are distinct edges of G incident with a common vertex then e1 and e2 are
adjacent edges.
Definition 1.2 The cardinality of the vertex set of a graph G is called the
order of G and is denoted by p. , The cardinality of its edge set is called the size
of G and is denoted by q. A graph with p vertices and q edges is called a (p,q) graph.
Definition 1.3 A graph H is called a subgraph of a graph G if' V(11) c V(G)
and E(H) c E(G). A spanning subgraph of G is a subgraph H with
V(H)=V(G).
Definition 1.4 The degree of a vertex v in a graph G is the number of edges
of G incident with v and is denoted by deg i'. A vertex of degree 0 is called an
isolated vertex and a vertex of degree 1 is called a pendant vertex. Any vertex
2
which is adjacent to a pendant vertex is called a support. The minimum and
maximum degrees of vertices of G are denoted by 6 and A respectively.
Definition 1.5 A graph is regular of degree r if every vertex of G has
degree r. Such graphs are called r- regular graphs.
Definition 1.6 A graph G is complete if every two of its vertices are adjacent.
A complete graph on p vertices is denoted by K.
Definition 1.7 A bipartite graph G is a graph whose vertex set V can be
partitioned into two subsets V1 and V2 such that every edge of G joins a vertex of
V1 and a vertex of V2 . If G contains every edge uv where u E V1 and v E V2 then
G is called a complete bipartite graph and is denoted by K,,,,,, where
Vi I = m and I
V2
I = n. The complete bipartite graph K1, ,, is called a star.
Definition 1.8 Let u and v he vertices of a graph G. A u-v walk oF G is a
finite alternating sequence u = uo, e1 , u1 , e2,..., u,,_ 1 , e,,,
edges such that e1
U- 1
u1 for i
U,, =
v of vertices and
1, 2, ..., n. The number ii is called the
length of the walk.
A walk u0, e1 , u1 ,..., u,, - ' e,
U 0,
u 1 , u2,.;.,
U, - 1, U n
U,, is
determined by the sequence
of its vertices and hence we specify a walk simply by
(u0, ti1 , ..., u,j. A walk in which all the vertices are distinct is called a path. A
walk (u0 , u1 , ..., u,,) is called a closed walk if u0 = u A closed walk in
which u0, U 1 , u2,..., u,,.. are distinct is called a cycle.
A path of length p is denoted by PP and a cycle on p vertices
is denoted by C.
3
Definition 1.9 A graph G is said to be connected if any two vertices of G are
joined by a path.
Definition 1.10 Let G 1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs. Then
their cartesian product G1 x G2 is defined to be the graph whose vertex set is
V1 xV2
and
u1u2
and edge
E
set is{(ul ,vi )(u2,v2)
u1 = u2 andv1 v2 EE2 orv1 =
V2
E, }.
Definition .!.!! Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs. Then
their join G1
E1
U
E2
u{
+
G 2 is the graph whose vertex set is Vi
U
V2
and edge set is
uv I u E Vi and V E V2 }.
Definition 1.12 The graph Ci,, + K1 is called a wheel and is denoted by W,,.
Definition 1.13 The graph P, + K1 , n ^! 2,is called afan and is denoted byJ1.
Definition 1.14 [21] A triangular snake or A - snake is a connected graph in
which each block is a triangle and the block-cutpoint graph is a path. We call a
snake with n blocks a A , - snake.
Definition 1.15 [11] A quadrilateral snake is a connected graph in which each
block is a quadrilateral and the block-cutpoint graph is a path.
Definition 1.16 A connected graph having no cycles is called a tree. A tree T is
called a caterpillar if removal of all its pendant vertices results in a path.
Definition 1.17 A chord of a cycle is an edge joining two otherwise
nonadjacent vertices of the cycle.
Definition 1.18 [10] A chord of a cycle C,, is called Pk - chord if it divides the
cycle into two cycles Ck and C,, - k
Definition 1.19 [1] Let G = (V, E) be a (p, q) graph. G is said to be strongly
indexable if there exists a bijection f : V—* 1 0, 1, 2, ...,p— 1} such that
f( E) = 11, 2, ..., q } where f(uv) = f( u) +f( v) for any edge uv E E
and f is called a strong indexer of G. G is said to he indexable if f : E -4 N is
injective and f is called an indexer of G. G is said to be strongly k-indexable if
f(E)= { k, k+ 1,..., k+q — 1 } andf is called a strong k-indexer of G.
Lemma 1.20
[1] For any strongly indexable graph q < 2p - 3
Theorem 1.21 [1] Every strongly indexable graph has exacty one non-trivial
component which is either a star or has a triangle.
Theorem 1.22 [1] If G is a strongly indexable graph with a triangle then any
strong indexer of G must assign 0 to a point of a triangle in G.
Definition 1.23 [3] Let G ( V, E) be a connected (1), q) graph. G is said to
be graceful if there exists an injection f: V -> 1 0, 1, 2, ..., q } such that when
each edge uv is labeled withI f( u) -f( v) I ' the resulting edge labels are
distinct.
Definition 1.24 [13] A Skoleni—graceful labeling of G is defined to be a
bijection
f: V -^ { 1, 2, ..., I V }
f*:E_{12
such
that
the
induced
labeling
IEI} defined byf*(uv)= If( u ) — f( v ) I is also a
bijection. Such anf is called aS - labeling of G and if it exists, G is then said to
be Skolem-graceful.
Definition 1.25 [11] A (p, q) - graph G= ( V, E) is said to be odd graceful
if there exists an injection
f: V -+ { 0 , 1 , 2 , ..., 2q - 1 }
such that
5
f*( E) = { 1, 3, ..., 2q - 1) where I *(u v) = I f(u) —f(v)
UV E
br any edge
E.
Definition 1.26 [24] A labeling of a graph G = (V, E) is a mapping f: V -4 N
where N is the set of all non-negative integers. For any edge e = UV E E we
define f
*()
= I f(u) -f( v) I . For a labeling
f: V( G) - {O, 1, 2, ... , k - 11 N, let v1 ( i) and e1 ( i) denote the number
of vertices and edges respectively with the label i. The labeling f is called vertex
k-equitable if I v1 ( i ) - v1 (j) I < 1 for all i , f E { 0 , 1 , 2, ..., k - 11. The
labeling f is called edge k-equitable
i ,j E
if I e1 ( i ) - e1 (j) J
I for all
{O, 1 , 2, ..., k— 1}. f is called k-equitable if it is both edge and vertex
k-equitable A graph G is called Ic-equitable if it admils a k-cquibal)Ic labeling.
Definition 1.27 [17] Let G (V, E) be a (p, q) graph.
Let
f: E - 11, 2, ..., q} be a bijection. For each vertex
V
in V define
f (v) the sum of the labels of the edges incident at v. If (v) is same for all
the vertices thenf is called a magic labeling and G is said to be magic if it admits
a magic labeling. If for any two distinct vertices u and v , f(u) # f(v) thenf is
said to be an antimagic labeling. A graph G is said to be antimagic if, it admits
an antimagic labeling.
Definition 1.28 [17] Let D = (V, A) be a directed graph of order p and size q.
Let!: A -* { 1, 2, ..., q } be a bijection. For any vertex v, let f( v) = the
sum of the labels of the arcs with
V
as head amid[( v ) = SUni of the labels of the
arcs with v as tail. If f( v) = f( v) for every vertex v then f is said to be a
conservative labeling and D is said to be a conservative if it admits such a
labeling.
Definition 1.29 [17] A graph G is said to be conservative if there exists an
orientation of G such that the resulting directed graph is conservative.
Definition 1.30 [10] A graph G is said to be D-magic if there exists an
orientation of G and a labeling f of the arcs of G with 1,2,..., q such that
If (u)—f (u) I = If( v )—f( v )
J
foranytwoverticesuandv.
Definition 1.31 [10] A graph G is said to be D-antiniagic if there exists an
orientation of G and a labeling f of the arcs of G with 1,2,.. . ,q such that
I f( u ) —f( u)
f(
v)—f -( v) I forany two distinct vertices u and v
Theorem 1.32 [8] C4 U Pm is graceful for all in ^ 3.
Conjecture 1.33 [8]
C',1 U
Pm is graceful if n = 3 or
n^
5 and ii + in ^: 7.
Frucht and Salinas [8] checked the truth of this conjecture for all combinations of
n and in that satisfy 7 :^ n + in < 16.
Theorem 1.34. [9] C,, U P,, is graceful for n = 3 or ii = 2 m + I with in ^! 3.
Notation 1.35 Let x be any real number. Then
LXJ
stands for the largest
integer less than or equal to x and Exl stands for the smallest integer greater
than or equal to x.
Notation 1.36. The graph obtained by identifying a vertex of C,, with th e
centre of the star K 1,11, is denoted by C,, OK1111.