### ALPHA LABELINGS OF DISJOINT UNION OF HAIRY CYCLES

#### Abstract

In this paper, we prove the following results: 1) the disjoint union of \(n\geq 2\) isomorphic copies of the graph which is obtained by adding a pendent edge to each vertices of the cycle of order 4 admits \(\alpha\)-valuation; 2) the disjoint union of two isomorphic copies of the graph which is obtained by adding \(n\geq 1\) pendent edge to each vertices of the cycle of order 4 is admits \(\alpha\)-valuation; 3) the disjoint union of two isomorphic copies of the graph obtained by adding a pendent edge to each vertex of the cycle of order \(4m\) admits \(\alpha\)-valuation; 4) the disjoint union of two non-isomorphic copies of the graph obtained by adding a pendent edge to each vertices of the cycle of order \(4m\) and \(4m-2\) admits \(\alpha\)-valuation; 5) the disjoint union of two isomorphic copies of the graph which is obtained by adding a pendant edge to each vertex of the cycle of order \(4m-1(4m+2)\) is admitted graceful (\(\alpha\)-valuation).

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J. Abrham and A. Kotzig, Graceful valuations of 2-regular graphs with two components, Discrete Math., 150 (1996), 3-15. R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory, Second Edition, Springer, 2009. C. Barrientos , Equitable labeling of corona graphs, J. Combin. Math. Combin. Comput., 41, (2002), 139-149. C. Barrientos, Graceful graphs with pendant edges, Australasian Journal Of Combinatorics, 33(2005), 99-107. C. Barrientos and S. Minion, Constructing Graceful graphs with Caterpillars, Journal of Algorithms and Computation, 48(2016), 117-125. M. Buratti, G. Rinaldi, and T. Traetta, Some results on 1-rotational Hamiltonian cycle systems, J. Combin. Designs, 22(6) (2014), 231-251. K. Eshghi and M. Carter, Construction of $\alpha$-valuations of special classes of 2-regular graphs, Topics in Applied and Theoretical Mathematics and Computer Science, Math. Comput. Sci. Eng., WSEAS, Athens (2001), 139-154. R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4(3), 322-325 (1970). R. Frucht and L.C. Salinas, Graceful numbering of snakes with constraints on the first label, Ars Comin., 20(1985), B, 143-157. J.A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, (2019), $\#DS6$ W. Golomb, How to number a graph, in Graph Theory and Computing, R. C. Read, ed., Academic Press, New York (1972), 23-37. A. Graf, A new graceful labeling for pendant graphs. Aequat. Math. 87(12),(2014), 135-145. R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Discrete Methods, 1 (1980), 382-404. A. Kumar, Debdas Mishra, A. Kumar and V. Kumar Alpha Labeling of Cyclic Graphs, Int. J. Appl. Comput., Math (2021), 7:151. A. Kumar, D. Mishra, A. Verma and V.K. Srivastava, Alpha Labeling of Cyclic Graphs - I, ARS Combinatorics, 154 (2021), 257-263. D.R. Lashmi and S. Vangipuram, An α-valuation of quadratic graph $Q(4, 4k),$ Proc. Nat. Acad. Sci. India Sec. A, 57 (1987), 576-580. S. Minion and C. Barrientos, Three Graceful Operations, Journal of Algorithms and Computation, 45 (2014), 13-24. P. Pradhan, A. Kumar, and D. Mishra, On Gracefulness of Graphs Obtained from Hairy Cycles, Journal of Combinatorics, Informations and System Sciences, 35 (2010), 471-480. P. Pradhan and A. Kumar, Graceful hairy cycles with pendent edges and some properties of cycles and cycle related graphs, Bull. Cal. Math. Soc., 104(1), (2012), 61-76. P. Pradhan and K. Kumar, On Graceful Labeling of Some Graphs with Pendant Edges, Gen. Math. Notes, 23, No. 2, August 2014, 51-62. P. Pradhan and K. Kumar, On $k$-Graceful Labeling Of Some Graphs, J. Appl. Math. and Informatics, 34(2016), No. 1 - 2, 9 - 17. G. Ringel, Problem 25, in Theory of Graphs and its Applications, Proc. Symposium Smolenice 1963, Prague (1964), 162. D. Ropp, Graceful labelings of cycles and prisms with pendant points, Congr. Number., 75(1990), 218-234. A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), 349-355. M. Truszczyn´ki, Graceful unicyclic graphs, Demonstratio Math., 17,(1984), 377-387.

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