Abstract
The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called k-shifted-antimagic labelings which allow the consecutive numbers starting from \(k+1\), instead of starting from 1, where k can be any integer. This paper establishes connections among various concepts proposed in the literature of antimagic labelings and extends previous results in three aspects:
-
Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be k-shifted-antimagic for sufficiently large k.
-
Some graphs are proved k-shifted-antimagic for all k, while some are proved not for some particular k.
-
Disconnected graphs are also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon, N., Kaplan, G., Lev, A., Roditty, Y., Yuster, R.: Dense graphs are antimagic. J. Gr. Theory 47, 297–309 (2004)
Chang, F.H., Liang, Y.C., Pan, Z., Zhu, X.: Antimagic labeling of regular graphs. J. Gr. Theory 82, 339–349 (2016)
Chang, F.H., Chin, P., Li, W.T., Pan, Z.: The strongly antimagic labelings of double spiders. Indian J. Discrete Math. 6(1), 43–68 (2020)
Cranston, D.W.: Regular bipartite graphs are antimagic. J. Gr. Theory 60, 173–182 (2009)
Cranston, D.W., Liang, Y.C., Zhu, X.: Regular graphs of odd degree are antimagic. J. Gr. Theory 80, 28–33 (2015)
Hartsfield, N., Ringel, G.: Pearls in Graph Theory, pp. 108–109. Academic Press, INC., Boston (1994). (Revised version)
Hefetz, D.: Anti-magic graphs via the combinatorial nullstellensatz. J. Gr. Theory 50, 263–272 (2005)
Huang, T.Y.: Antimagic Labeling on Spiders. Master Thesis, Department of Mathematics, National Taiwan University (2015)
Kaplan, G., Lev, A., Roditty, Y.: On zero-sum partitions and antimagic trees. Discrete Math. 309, 2010–2014 (2009)
Lee, M.J., Lin, C., Tsai, W.H.: On antimagic labeling for power of cycles. Ars Comb. 98, 161–165 (2011)
Lozano, A., Moray, M., Seara, C.: Antimagic labelings of caterpillars. Appl. Math. Comput. 347, 734–740 (2019)
Liang, Y.C., Zhu, X.: Anti-magic labeling of cubic graphs. J. Gr. Theory 75, 31–36 (2014)
Liang, Y.C., Wong, T., Zhu, X.: Anti-magic labeling of trees. Discrete Math. 331, 9–14 (2014)
Shang, J.L.: Spiders are antimagic. Ars Comb. 118, 367–372 (2015)
Shang, J.L.: $P_2, P_3, P_4$-free linear forests are antimagic. Util. Math. 101, 13–22 (2016)
Shang, J.L., Lin, C., Liaw, S.C.: On the antimagic labeling of star forests. Util. Math. 97, 373–385 (2015)
Wang, T.M.: Toroidal grids are anti-magic. Lect. Notes Comput. Sci. 3595, 671–679 (2005)
Wang, T.M., Hsiao, C.C.: On anti-magic labeling for graph products. Discrete Math. 308, 3624–3633 (2008)
Wong, T., Zhu, X.: Antimagic labeling of vertex weighted graphs. J. Gr. Theory 70, 34–350 (2012)
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.
F.-H. Chang: supported by MOST 106-2115-M-003-005. H.-B. Chen: supported by MOST 105-2115-M-035-006-MY2. W.-T. Li: supported by MOS T105-2115-M-005-003-MY2.
Rights and permissions
About this article
Cite this article
Chang, FH., Chen, HB., Li, WT. et al. Shifted-Antimagic Labelings for Graphs. Graphs and Combinatorics 37, 1065–1082 (2021). https://doi.org/10.1007/s00373-021-02305-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-021-02305-w