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INTRODUCTION

One of the most interesting problems in the area of Graph

Theory is that of labeling of graphs. A labeling or valuation or numbering

of a simple graph G is an one-to-one mapping from its vertex set into a

set of non-negative integers which induces an assignment of labels to the

edges of G.

Labeled graphs serve as useful models for a broad range of

applications. They are useful in many coding theory problems, including

the design of good radar type codes [25], Synch - set codes, missile

guidance codes and convolutional codes with optimal non-standard

encodings of integers. Labeled graphs have also been applied in

determining ambiguities in X-ray crystallographic analysis, and designing

a communication network addressing system in determining optional

circuit layouts and radio astronomy problems.

The impatus for graph labeling problem began with the

conjecture of G. Ringel [101] which states that "All trees are graceful".

In 1967 Rosa [99] defined ~-valuations of G and Golomb

[59] called such labelings graceful. Much work was done with graceful

labeling. A large number of families has been numbered gracefully. See

{ [1], [2], [3], [5], [10], [13], [21], [25], [26], [27], [28], [32], [33], [34], [38],

[42], [44], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [58],

[59], [66], [69], [72], [74], [76], [77], [78], [79], [80], [82], [88], [89], [92],

[93], [96], [98], [99], [100], [101], [104], [107], [108], [109], [110], [111],

[112], [115], [116], [117], [118]}.

Apart from the graceful labeling, some other labelings were

defined and developed. Graham and Sloane [61] introduced harmonious

graphs {see [24], [41], [54], [55], [56], [57], [58], [61], [90], [102], [103] }.

Some of the other known labelings are sequential, Arithmetic { [4], [6], [8],

[23], [37], [60], [67], [71], [108], [109], [114], [115], [119], [120], [121] }

indexable labeling { [9], [12], [22] }. Edge graceful labeling { [91], [94],

[95] } Sum labeling { [11], [35], [64] }, Skolem graceful { [16], [43], [87] },

Felicitous labeling { [22], [86], [106] }, set sequential labeling [7], Magic

labeling { [15], [17], [18], [19], [20] }

The thesis, II A Study on different classes of graphs and

their labelings", proposes a study of different types of labelings and

various classes of graphs that possess such labelings.

This thesis consists of five chapters. Special emphasis has

been given to Triangular gracefulgraphs, Antimagic graphs, Consecutive

graphs and E-cordial graphs.

In chapter I, we give a brief survey of graph labelings and

we collect some basic definitions and results which are useful for the

subsequent chapters. For basic graph theoretic terminologies,

definitions and notations we follow { [31], [63], [97] }.

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Chapter II concentrates on Triangular graceful graphs. We

.[48] introduced triangular graceful graphs. In Number Theory [113]

triangular numbers are defined.

In this chapter we prove the following graphs, are triangular

graceful. Path Pm, olive trees, caterpillars, the star Sk,m, the one point

union of k copies of path of length m, cycles Cn, n == 0 (mod 4), We also

prove certain classes of graphs such as wheels, Kn, Km n, m,n ;t:. 1, are,

not triangular graceful. We set forth the conjecture that all trees are

triangular graceful.

Chapter III deals with Antimagic graphs. In this chapter we

prove the following graphs are Antimagic.

1. Crown Cn 0 K1

2. Cn 0 Pm [for m =nand m =n + 1 ]

3. The double crown Cn 0 P2 \;j odd n ~ 3

4. < Cn, t > I the one point union of t copies of Cn

5. The grid Cn x Cn for even n

6. Pn + K t

7. Pn X C3 for all even n

8. Triangular snakes, quadrilateral snakes, K4 snakes

9. K1 n + K t

10. Book graph Bn

In Chapter IV, we study the consecutive labeling. In this

chapter we prove the following graphs are consecutive graphs.

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1. The prism Cn X P2 has magic vertex labeling, and edge

consecutive labeling.

2. The n-gonal snakes are edge consecutive.

3. Quadrilateral and triangular snakes are edge consecutive

4. The graph Cn 11 Cm is edge consecutive.

5. < Cn m >, the one point union of m copies of cycle Cn, has,

consecutive labeling.

Also we construct some classes of graphs and prove they

are consecutive.

In chapter V we consider E-cordial graphs. In this chapter

we prove the following graphs are E-cordial.

1. K2 0 Cn, V n :f. 0 (mod 2)

2. Cn 0 P3 V even n

3. Closed helm W*(t,n) for t = 2,4,6

4. The graph L1Cn, Lotus inside a circle Vn,

5. Book graph Bn when n is odd

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6. Mirror sum graph M(m,n) when m + n == 0 (mod 2)

7. The generalised Petersen graph P(n,k) for all k and even n.

8. The graph Hn,n with vertex set {u1,u2, ... ,un,v1,v2, ... ,vn} and the edge

set {UjVj 11 ~ i ~ j ~ n } is E-cordial for n == 0 (mod 4)

10. Ladder Ln = Pn X K2

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