Abstract
One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261–290, 1973) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star.
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References
Aldred R.E.L., McKay B.D.: Graceful and harmonious labellings of trees. Bull. Inst. Comb. Appl. 23, 69–72 (1998)
Beardon A.F.: Magic labellings of infinite graphs. Austral J. Comb. 30, 117–132 (2004)
Cavenagh N., Combe D., Nelson A.M.: Edge-magic group labellings of countable graphs. Electron J. Comb. 13, #R92 (2006)
Chan T.L., Cheung W.S., Ng T.W.: Graceful tree conjecture for infinite trees. Electron J. Comb. 16, #R65 (2009)
Combe D., Nelson A.M.: Magic labellings of infinite graphs over infinite groups. Austral J. Comb. 35, 193–210 (2006)
Edwards M., Howard L.: A survey of graceful trees. Atlantic Electron J. Math. 1, 5–30 (2006)
Fulman J., Hanson D., MacGillivray G.: Vertex domination-critical graphs. Networks 25, 41–43 (1995)
Gallian J.A.: A dynamic survey of graph labeling. Electron J. Comb. 5, #DS6 (2005)
Hrnčiar P., Haviar A.: All trees of diameter five are graceful. Discret. Math. 233, 133–150 (2001)
Huang C., Kotzig A., Rosa A.: Further results on tree labellings. Util. Math. 21c, 31–48 (1982)
Kotzig A.: On certain vertex-valuations of finite graphs. Util. Math. 4, 261–290 (1973)
Rosa, A.: 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, 349-355 (1967)
Stanton, R.A., Zarnke, C.R.: Labelling of balanced trees. Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1973). Utilitas Math., 479-495 (1973)
Traetta T.: A complete solution to the two-table Oberwolfach problems. J. Comb. Theory Ser. A 120(5), 984–997 (2013)
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Chan, T.L., Cheung, W.S. & Ng, T.W. Graceful labeling for mushroom trees. Aequat. Math. 89, 719–724 (2015). https://doi.org/10.1007/s00010-014-0259-5
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DOI: https://doi.org/10.1007/s00010-014-0259-5