AbstractA (p, q)-graph G = (V, E) is said to be strongly k-indexable if it admits a strong k-indexer viz., an injective function ƒ:V→{0, 1, 2, …, p − 1} such that ƒ(x)+ƒ(y)=ƒ+(xy)ϵ⨍+(E)={k, k + 1, k + 2, …, k + q − 1}.In the terms defined here, k will be omitted if it happens to be unity. We find that a strongly indexable graph has exactly one nontrivial component which is either a star or has a traingle. In any strongly k-indexable graph the minimum point degree is at most 3. Using this fact we show that there are exactly three strongly indexable regular graphs, viz. K2, K3 and K2xK3. If an eulerian (p, q)-graph is strongly indexable then q ϵ 0, 3(mod4)

Acharya, B.D.

Hegde, S.M.

Elsevier - Publisher Connector 

Strongly indexable graphs 

Discrete Mathematics 93 (1991) 123-129 
North-Holland 
123 
Acharya 
Department of Science & Technology, Gouernment of India, Technology Bhavan, New 
Mehrauli Road, New Dehli-110016, India 
S.M. Hegde 
Department of Mathematics, University of Delhi, Delhi-l 10007, India 
Received 25 August 1989 
Abstract 
Acharya, B.D. and S.M. Hegde, Strongly indexable graphs, Discrete Mathematics 93 (1991) 
123-129. 
A (p, q)-graph G = (V, E) is said to be strongly k-indexable if it admits a strong k-indexer viz., 
an injective function f : V - {C, 1, 2, . . . , p - 1) such that 
f(x)+f(y)=f+(xy)Ef+(E)={k,k+l,k+2,.. . , k+q-1). 
In the terms defined here, k will be omitted if it happens to be unity. We find that a strongly 
indexable graph has exactly one nontrivial component which is either a star or has a traingle. In 
any strongly k-indexable graph the minimum point degree is at most 3. Using this fact we show 
that there are exactly three strongly indexable regular graphs, viz. &, K, and K, X K,. If an 
eulerian (p, q)-graph is strongly indexable then 9 = 0, 3 (mod 4). 
Unless mentioned otherwise, we consider only finite simple graphs as treated in 
PI . 
Labeling of the points of a given graph G = (V, E) by integers is merely an 
assignment of certain distinct nonnegative integers to the points of G. In other 
words, it is simply an injective function f : V + A, from the point set V = V(G) of 
G into a subset A of the set N of nonnegative integers. 
Several practical problems in real-life situations have motivated the study of 
labelings of graphs which are required to obey a variety of conditions depending 
on the structure of graphs. There is an enormous amount of literature built up on 
several kinds of labelings over the past two decades or so. It would be too 
unwieldly to cite all references, but a few pioneering papers like [l-3,5-6, lo-111 
are vvorth comprehension. 
In this paper, we are interested to investigate the properties of (p, &-graphs 
G = (V, E j which admit a labeling f : V-+ (0, 1,2, . . . , p - 1) such that the 
values f’(e) of the lines e = uv of 6, defined as f(u) -t-f(v), are all distinct-that 
0012-365X/91/$03.50 @ 1991 s Elsevier Sciencs Publishers B.V. All rights reserved 
124 B. D. Acharya, S.M. Hegde 
is, such that the so induced function f’ : E(G)+ N is injective too-such a 
labeling f of G being called an i&exer of G. We shall call a graph indexable if it 
admits an indexer and by an indexed graph we shall mean an indexable graph 
together with an indexer. 
For any labeling f of a given graph G, we shall write 
f(G) = If(u): u E V(G)h f ‘I@ = {f+(e): e E E(G)), 
and fmax(G) and f&,,(G) will denote the maximum values in f(G) and f+(G) 
respectively. 
Observation 1. For any indexer f of a given (p, q)-graph G, since f(G) = 
(0, 1, 2, . . . ,p-1) wehavef+(G)s{1,2 ,... ,2p-3). 
Hence, for any indexable (p, q)-graph G we must have 
qS2p-3. (1) 
The bound in (1) is sharp. For example, for any integer p 2 3, consider the join 
G = K2 -I- I&. Let the points of K2 be labeled u1 and up, and those of &,_2 be 
labeled u2, u3, . . . , u,,+ Then the mapf : V(G)+ (0, 1,2, . . . , p - 1) defined 
bY 
f (Uj)= i - 1, 1SiSp (2) 
can be easily verified to be an indexer of G. 
This motivates the following new notion. A (p, q)-graph G = (V, E) is said to 
be strongly indexable if it admits a strong indexer, viz, an indexer f such that 
f+(G)= {1,2,. . . , q ). In general, G is said to be strongly k-indexable if it 
admits a strong k-indexer, that is, an indexer f so that f+(G) = {k, k + 1, k + 
2 9 - * - 9 k + q - l} (see Fig. 1). 
Before we proceed with our investigations on these new notions, a few words 
about their relation to certain other known graph labeling problems are worth 
mentioning here. 
Grace [7-81 calls a (p, q)-graph G = (V, E) sequential if it admits a sequentiaz 
labeling of characteristic k, viz., a labeling f : V+ (0, 1, 2, . . . , M) such that 
f+(G)=(k,k+l,k+2 ,..., k+q-1) where M=q if G is a tree and M= 
q - 1 if G is not a tree. Clearly, by this definition it follows that the notions of 
sequential labelings of characteristic k and strong k-indexers agree on the classes 
of trees and unicyclic graphs. 
Chang et al. [4] define a (p, q)-graph G = (V, E) to be strongly k-elegant if it 
admits a strong k-elegant labeling which is a labeling f : V-, (0, 1, 2, . . . , q} of G 
such that f’(G) = (k, k + 1, k + 2, . . . , k + q - 1). Here too, we observe that 
the notions of strong k-elegant labelings and strcmg k-indexers agree on the class 
of trees. 
Strongly indexable graphs 125 
A STRONGLY INDEXED GRAPH 
2* 6 
A STRONGLY 3 - INDEXED GRAPH 
Fig. 1. 
Chang et al. [4] also define a (p, q)-graph G = (V, E) to be strongly 
k-harmonious if it admits a strong k-harmonious labeling which is a labeling 
f:V+(O, 1,2,. . . ,q-1) such that f+(G)={k,k+l,k+2,...,k+q-1). 
It is easily seen that the notions of strongly k-harmonious graphs and strongly 
k-indexable graphs agree on the class of unicyclic graphs. 
Some examples of non-indexable graphs are provided by the following result. 
Theorem 1. The complete graph Kn is indexable if and only if n s 3. 
The following 
graphs. 
result sheds some light on the structure of strongly indexable 
Theorem 2. Every strortgly inaTexable graph has exactly one nontrivial component 
which is either a star or has a triangle. 
Proof. It is enough to show that a strongly indexable triangle-free graph (i.e., a 
graph without triangles) is a forest having exactly one nontrivial component which 
is a star. Towards this end, let G = (V, E) be a triangle-free (p, q)-graph having a 
strong indexer J Let u E V be such that f(u) = 0. Since f is a strong indexer, 
1 ef+(G) so that there exists a point v1 in the neighbourhood N(u) = {v E 
V: uv E E} of u with f(v,) = 1. Similarly, 2 ef+(G) and since f is injective one 
can easily check that there exists a point v2 E N(u) such that f(v2) = 2. Let t be 
126 B.D. Acharya, SM. Hegde 
the largest integer in f+(G) such that 1,2,3,. . . , t - 1 ef+(E,(G)), where 
E,(G) denotes the set of lines of G that are incident at u, and let f+(xy) = t. 
Since t $f*(EJG)) we must have O<f(x) <t, 0 <f(y) < t and t =f+(xy) = 
f(x) +f(y) so that X, y E N(u). But then it follows that u, X, y is a triangle in G, a 
contradiction to our supposition that G is triangle-free. Cl 
Thus, the class of connected triangle-free graphs which are not stars is an 
infinite class of graphs which are not strongly indexable. This class contains, for 
example, all bipartitie graphs other than stars, all cycles Cn, n 2 4, etc. 
Corollary 2.1. For the star Kl,p--l there is a unique strong indexer, viz., the one 
which assigns 0 to the central point and the numbers 1, 2, . . . , p - 1 to the points 
of unit degree. 
Corollary 2.2. 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. 
Remark 1. Hence, if a unicyclic graph G is strongly indexable then its unique 
cycle must be a triangle. Let U3 denote the class of unicyclic graphs in each of 
which the unique cycle is a triangle. For any integer d 3 1, we have a construction 
of a strongly indexable graph of diameter d in US. However, to determine 
precisely which graphs in US are strongly indexable is an open problem. 
Conjecture 1. All unicyclic graphs are indexable. 
In support of this conjecture we have been able to establish that all cycles are 
indexable. 
For any (p, q)-graph G = (V, E) and for any function f : V-, N, the identity 
c d(u)f (u) = c f’(e) 
UEV(G) cd(G) 
(3) 
can be proved by easy counting arguments. 
Given a labeling f : V(G)+ (0, 1,2, . . . , p - l} of a (p, q)-graph G = (V, E), 
we may have a canonical labeling of the points of G so that a point u with f(u) = i 
is labeled ui, 0 G i up - 1. If G has a strong 
labeling of G, with respect to f, we may derive 
and hence if f is a strong indexer of G then (4) 
k-indexer f then for a canonical 
from (3) the identity 
(4) 
gives us 
‘2’ i l d(ui) = q(q + 1)/2. 
i=l 
(5) 
Identity (4) will be used in the proof of the following result. 
Strongly indexable graphs 127 
Theorem 3. The minimum point degree 6 of any strongly k-indexable graph with 
at least two points is at most 3. 
Proof. Let G = (V, E) be a (p, @-graph, p a 2, having a strong k-indexer fi 
Then using the canonical labeling of the points of G with respect to f and (4) WI= 
find that 
(q2 + (2k - l)q)/2 ~~2’ i l d(ui) 2 6 “2’ i = Sp(p - 1)/z; 
i=O i=O 
that is, 
q2 + (2k - 1)q 3 Sp(p - 1). 
By Observation 1 we have q + k - 1 s 2p - 3 which amounts to saying 
(6) 
q<2(p-1)-k (7) 
for any strongly k-indexable graph. Hence, using (7) in (6) we find 
[2(p - 1) - k12 + (2k - 1)[2(p - 1) - k) 3 Sp(p - 1) 
which, on simplifying, yields 
4p2- lop-k2+k+6HSp(p-1). (8) 
Now, if 6 > 3, since 6 is an integer we have S 2 4 whence (8) yields 
-6(p-l)-k(k-1)sO. 
This inequality is never possible for p 2 2 and k 2 1. Cl 
Corollary 3.1. For any strongly k-indexable graph G, the connectivity k(G), the 
line connectivity A(G) and S(G) stand in the relation k(G) s A(G) s 6(G) s 3. 
The following result determines the regular strongly indexable graphs. 
Corollary 3.2. There are exactly three nontrivial regular graphs which are strongly 
indexable, viz., K2, K3 and K2 x KS. 
The following result gives a special necessary condition for an eulerian graph to 
be strongly indexable. 
Theorem 4. If an eulerian (p, q)-graph is strongly indexable then 
q = 0,3 (mod 4). 
Proof. Since the degree of every point of an eulerian graph is even this result 
follows from (5). Cl 
S. M. He& 
is a strongly indexable eulerian graph. 
counterexamples to the converse of 
128 B. D. Acharya, 
For example, K2 + & for odd p b 3 
The cyc!es Cn ,3 < n = 0, 3 (mod 4) are 
Theorem 4. 
, 
The following is a general method of constructing new strongly indexable 
graphs from an arbitrarily given strongly indexable graph. 
Let G = (V, E) be any (p, &graph having a strong indexer f, and let 
G’ = (V’, E’) be a (p’, &)-graph together with a strong indexer f’ such that 
p’ 3 (2p - 4). Define a function f” on G’ by 
f”(u) =: f ‘(u) + p for every u E V’. 
We then construct another graph H = (V U V’, E U E’ U F) where F is a set of 
new lines defined by joining the point K of G with f (u) = q - p + 1 to the points IJ 
of G’ with f’(v)=p+i, 0+<2p-q-1. Since 2p-q-1=2p-(q+l)s 
2p - 1 we see that this construction is well-defined. Then one can easily see that 
the function g = f U f’ is a strong indexer of H. This construction is illustrated in 
Fig. 2(b) for the connected ‘maximal’ strongly indexed graph (i.e., with 
q = 2p - 3) of Fig. 2(a). 
For any finite set X, a partition (X,, X,} of X is said to be central if the 
cardinalities of X1 and X2 differ by at most unity. By a partition of a graph G we 
mean actually a partition of the point set V(G) of G. 
Tiieorem 5. if a (p, q j-graph G = (V, Ej is strongly k-indexable then G has a 
central partition ( VO, Ve> such that exactly [q/21 lines of G have their both ena5 
not belo,nging to the same subset Va, a = 0, e, where [ 1 denotes the least integer 
function. 
---__ 
!!D! 
Fig. 2. 
Strongly indexable graphs 129 
The following theorem brings forth an interesting and useful property of the 
off-diagonals of the standard adjacency matrix of a canonically labeled indexed 
graph. We were motivated in this direction by tht: matrix representation of 
sequential graphs propounded originally by Grace [75. 
Theorem 6. Let G = (V, E) be a (p, q)-graph together with an indexer f Let 
A(G) = (aii) denote the adjacency matrix of G with respect o the canonical abeling 
of the points of G obtained from f= Thn=- a .cGh every offdiagonal of A(G) has either no 
nonzero entry or has exactly two l’s viz., those given by 
ati = llji =l wheref+(uiuj)=f+(Ujui)=f(ui)+f(uj)=i+j. 
Furthermore, if we label by i + j the off-diagonal containing the entries aij and aji, 
then f is a strong k-indexer of G if and only if the only nonzero off-diagonals of 
A(G)arelabeledk,k+l,k+2 ,..., k+q-1. 
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Strongly indexable graphs

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