Discussiones Mathematicae Graph Theory 29(3) (2009) 481-498
doi: 10.7151/dmgt.1459

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Lixia Fan and Zhihe Liang

Department of Mathematics, Hebei Normal University
Shijiazhuang 050016, P.R. China
e-mail: zhiheliang@163.com.cn


S.M. Lee proposed the conjecture: for any n > 1 and any permutation f in S(n), the permutation graph P(Pn,f) is graceful. For any integer n > 1 and permutation f in S(n), we discuss the gracefulness of the permutation graph P(Pn,f) if f = ∏k = 0l-1(m+2k,m+2k+1), and ∏k = 0l-1(m+4k,m+4k+2)(m+4k+1,m+4k+3) for any positive integers m and l.

Keywords: permutation graph; graceful, Lee's conjecture.

2000 Mathematics Subject Classification: 05C78.


[1] A. Rosa, On certain valuations of the vertices of a graph, in: Theory of Graphs, Proc. of Intemational Symposium, Rome 1966 (Gordon Breach, New York 1967), 349-355.
[2] S.W. Golomb, How to number a graph? in: R.C. Read, ed., Graph Theory and Computing (Academic Press, New York, 1972) 23-27.
[3] Z. Liang, On the graceful conjecture of permutation graphs of paths, Ars Combin. 91 (2009) 65-82.
[4] J.A. Gallian, A dynamic survey of graph labeling, Electronic J. Combin. 14 (2007) DS#6, 1-180.
[5] J.C. Bermond, Graceful graphs, radio antennae and Fench windmills, in: R.J. Wilson, ed., Graph Theory and Combinatorics (Pitman, London, 1979), 13-37.
[6] A.K. Dewdney, The search for an invisible ruler that will help radio astronomers to measure the earth, Scientific American, Dec. (1986) 16-19, doi: 10.1038/scientificamerican0186-16.
[7] S.M. Lee, K.Y. Lai, Y.S. Wang and M.K. Kiang, On the graceful permutation graphs conjecture, Congr. Numer. 103 (1994) 193-201.
[8] M. Maheo, Strongly graceful graphs, Discrete Math. 29 (1980) 39-46, doi: 10.1016/0012-365X(90)90285-P.
[9] C. Delorme, Two sets of graceful graphs, J. Graph Theory 4 (1980) 247-250, doi: 10.1002/jgt.3190040214.
[10] R.W. Frucht and J.A. Gallian, Labelling prisms, Ars Combin. 26 (1988) 69-82.
[11] J.A. Gallian, Labelling prisms and prism related graphs, Congress. Numer. 59 (1987) 89-100.
[12] Z. Liang, H. Zhang, N. Xu, S. Ye, Y. Fan and H. Ge, Gracefulness of five permutation graphs of paths, Utilitas Mathematica 72 (2007) 241-249.
[13] N. Han and Z. Liang, On the graceful permutation graphs conjecture, J. Discrete Math. Sci. & Cryptography 11 (2008) 501-526.

Received 13 November 2007
Revised 21 April 2009
Accepted 4 June 2009