Abstract
A graceful labeling of a graph G is an injective function from the vertex set of G to the set \(\{0,1,\dots ,|E(G)|\}\) such that the induced edge labels are all different, where an induced edge label is defined as the absolute value of the difference between the labels of its end vertices. If the induced edge labeling is simultaneously antimagic, i.e., the sums of labels of all edges incident to a given vertex are pairwise distinct for different vertices, we say that the graceful labeling is graceful antimagic. In this paper we deal with the problem of finding some classes of graceful antimagic graphs.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Aldred, R.E.L., McKay, B.D.: Graceful and harmonious labellings of trees. Bull. ICA 23, 69–72 (1998)
Aldred, R.E.L., Širáň, J., Širáň, M.: A note on the number of graceful labellings of paths. Discrete Math. 261, 27–30 (2003)
Arumugam, S., Premalatha, K., Bača, M., Semaničová-Feňovčíková, A.: Local antimagic vertex coloring of a graph. Graphs Combin. 33, 275–285 (2017)
Bača, M., Miller, M.: Super Edge-Antimagic Graphs. Brown Walker Press, Boca Raton (2008)
Bača, M., Miller, M., Phanalasy, O., Ryan, J., Semaničová-Feňovčíková, A., Sillasen, A.A.: Totally antimagic total graphs. Australasian J. Combin. 61(1), 42–56 (2015)
Bača, M., Miller, M., Ryan, J., Semaničová-Feňovčíková, A.: Magic and Antimagic Graphs, Attributes, Observations, and Challenges in Graph Labelings. Springer, Cham (2019)
Beutner, D., Harborth, H.: Graceful labelings of nearly complete graphs. Result. Math. 41, 34–39 (2002)
Exoo, G., Ling, A.C.H., McSorley, J.P., Phillips, N.C.K., Wallis, W.D.: Totally magic graphs. Discrete Math. 254(1–3), 103–113 (2002)
Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin. #DS6 (2017)
Golomb, S.W.: How to number a graph. Graph Theory and Computing, R.C. Read (ed.), Academic Press, pp. 23–37 (1972)
Hartsfield, N., Ringel, G.: Pearls in Graph Theory. Academic Press, Boston (1990)
Marr, A.M., Wallis, W.D.: Magic Graphs, 2nd edn. Birkhäuser, New York (2013)
Marzuki, C.C., Salman, A.N.M., Miller, M.: On the total irregularity strength of cycles and paths. Far East J. Math. Sci. 1, 1–21 (2013)
Rosa, A.: On certain valuations of the vertices of a graph. Theory of Graphs, International Symposium, ICC Rome 1966, Paris, Dunod, pp. 349–355 (1967)
Ryan, J., Munasinghe, B., Tanna, D.: Reflexive irregular labelings. Preprint
Sivaraman, R.: Graceful graphs and its applications. Int. J. Curr. Res. 8(11), 41062–41067 (2016)
Sloane, N.J.A.: A Handbook of Integer Sequences. Academic Press, New York (1973)
Acknowledgements
This work was supported by the Slovak Research and Development Agency under the contract No. APVV-19-0153 and by VEGA 1/0243/23.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmed, M.A., Semaničová-Feňovčíková, A., Bača, M. et al. On graceful antimagic graphs. Aequat. Math. 97, 13–30 (2023). https://doi.org/10.1007/s00010-022-00930-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-022-00930-1