Alpha labelings of disjoint union of hairy cycles

G. Rajasekaran   ; L. Uma

Ural mathematical journal, Tome 10 (2024) no. 1, pp. 123-135

Résumé

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

Détail

Keywords: Hairy cycles, Graceful valuation
Mots-clés : $\alpha$-valuation

Bibliographie

[1] Abrham J., Kotzig A., “Graceful valuations of 2-regular graphs with two components”, Discrete Math., 150:1–3 (1996), 3–15 | DOI | MR | Zbl

[2] Balakrishnan R., Ranganathan K., A Textbook of Graph Theory, 2nd, Springer, NY, 2012, 292 pp. | DOI | MR | Zbl

[3] Barrientos C., “Equitable labeling of corona graphs”, J. Comb. Math. Comb. Comput., 41 (2002), 139–149 | MR | Zbl

[4] Barrientos C., “Graceful graphs with pendant edges”, Australas. J. Comb., 33 (2005), 99–107 | MR | Zbl

[5] Barrientos C., Minion S., “Constructing graceful graphs with caterpillars”, J. Algorithms Comput., 48:1 (2016), 117–125 | DOI

[6] Eshghi K., Carter M., “Construction of $\alpha$-valuations of special classes of 2-regular graphs”, Math. Comput. Sci. Eng., 2001, 139–154 | MR | Zbl

[7] Frucht R., Harary F., “On the corona of two graphs”, Aequationes Math., 4 (1970), 322–325 | DOI | MR | Zbl

[8] Frucht R. W., Salinas L. C., “Graceful numbering of snakes with constraints on the first label”, Ars Comin., 20B (1985), 143–157 | MR | Zbl

[9] Gallian J. A., “A dynamic survey of graph labeling”, Electron. J. Combin., 2023, DS6, 1–644 | DOI | MR

[10] Graf A., “A new graceful labeling for pendant graphs”, Aequat. Math., 87 (2014), 135–145 | DOI | MR | Zbl

[11] Kumar J., Mishra D., Kumar A., Kumar V., “Alpha labeling of cyclic graphs”, Int. J. Appl. Comput., 7 (2021), 151, 1–7 | DOI | MR

[12] Kumar A, Mishra D, Verma A., Srivastava V. K., “Alpha labeling of cyclic graphs. I”, ARS Combinatorics, 154 (2021), 257–263 | MR | Zbl

[13] Lakshmi D. R., Vangipuram S., “An $\alpha$-valuation of quadratic graph $Q(4, 4k)$”, Proc. Nat. Acad. Sci., India, Sect. A, 57:4 (1987), 576–580 | MR

[14] Minion S., Barrientos C., “Three Graceful Operations”, J. Alg. Comp., 45:1 (2014), 13–24 | DOI

[15] Pradhan P., Kumar A., “Graceful hairy cycles with pendent edges and some properties of cycles and cycle related graphs”, Bull. Calcutta Math. Soc., 104 (2012), 61–76 | MR | Zbl

[16] Pradhan P., Kumar K., “On graceful labeling of some graphs with pendant edges”, Gen. Math. Notes, 23:2 (2014), 51–62 | MR

[17] Pradhan P., Kumar K., “On $k$-graceful labeling of some graphs”, J. Appl. Math. Inform., 34:1–2 (2016), 9–17 | DOI | MR | Zbl

[18] Pradhan P., Kumar A., Mishra D., “On gracefulness of graphs obtained from hairy cycles”, J. Combin. Inform. Syst. Sci., 35 (2010), 471–480 | MR

[19] Ringel G., “Problem 25”, Theory of Graphs and its Application. Proc. Symposium Smolenice, 1963, Prague, 1964 | MR

[20] Ropp D., “Graceful labelings of cycles and prisms with pendant points”, Congr. Number, 75 (1990), 218–234 | MR | Zbl

[21] Rosa A., “On certain valuations of the vertices of a graph”, Theory of Graphs: Int. Symposium, Rome, July 1966, Gordon and Breach, N.Y. and Dunod, Paris, 1967, 349–355 | MR

[22] Truszczyński M., “Graceful unicyclic graphs”, Demonstratio Math., 17 (1984), 377–387 | MR | Zbl