圖形分解是圖論中的一個非常重要課題。
因為它可以連接組合，代數和其他數學結構。
另一方面，圖分解的結果可以應用在編碼理論，實驗設計，計算機和通信網絡等領域。

反魔圖是圖形的一種標號。一般圖形標號是將圖形內的頂點
或邊給予對應的一個整數值的標號，
或兩者兼而有之。圖形標號推出在20世紀60年代中後期,
在這幾十年,圖形標號的研究論文已超過1500篇。
圖形標號結果已應用在計算機和通信網絡，應用統計學的研究，與一些設計科學等領域。

Graph decomposition is an important subject of graph theory.
Many combinatorial, algebraic, and other mathematical structures
are linked to decompositions of graphs,
which gives their study a great theoretical importance.
On the other hand, results on graph decompositions
can be applied in coding theory, design of experiments,
computer and communication networks, and other fields.
Nowadays, graph decomposition ranks the most prominent area
in graph theorey, even in combinatorics.
A graph is called an antimagic graph, if there exists an edge labeling which is an assignment of integers to the vertices
or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s.
In the intervening years dozens of graph labelings techniques have been studied in over 1500 papers.
Graph labeling can be applied in computer and communication networks, applied statistics research,
and some design science other fields.

[1] A. Rosa, On certain valuations of a graph, Theory of Graphs, Rome
(1966) 349-355
[2] B. Alspach, D. Dyes and D. L. Kreher, On isomorphic factorization of
circulant graphs, J. Combin. Des. 14 (2006) 406-414.
[3] B. W. Jackson, Some cycle decompositions of complete graphs, Journal
of Combinatorics, Information and System Sciences 13 (1988), 20{32.
[4] C. A. Parker, Complete bipartite graph path decompositions Ph. D. Dissertation,
Auburn University, Auburn, Alabama, 1998.
[5] C. Huang and A. Rosa, On the existence of balanced bipartite designs,
Utilitas Math. 4 (1973), 55{75.
[6] C. Huang, On the existence of balanced bipartite designs II, Discrete
Math 9 (1974), 147{159.
90
[7] C. Lin , J. J. Lin and Tay-woei Shyu, Isomorphic star decompositions of
multicrowns and the power of cycles, Ars Combin. 53 (1999), 249-256.
[8] C. Lin and T.-W. Shyu, A necessary and sucient condition for the star
decomposition of complete graphs, J. Graph Theory 23 (1996), 361{364.
[9] C. Vanden Eynden, Decomposition of complete bipartite graphs. Ars
Combin. 46 (1997), 287-296.
[10] D. De Caen and D. G. Homan, Impossibility of decomposing the complete
graph on n points into n􀀀1 isomorphic complete bipartite graphs,
SIAM J. Discrete. Math. 2 (1989), 48-50.
[11] D. Fron~cek, Note on cyclic decompositions of complete bipartite graphs
into cubes, The SeventhWorkshop "3 in 1" Graphs'98 (Krynica), Discuss.
Math. Graph Theory 19 (1999), no. 2, 219-227.
[12] D. Hefetz, Antimagic graphs via the combinatorial nullstellensatz, J.
Graph Theory 50 (2005) 263-272.
[13] D. Hefetz, T. Muutze, and J. Schwartz, On antimagic directed graphs,
J. Graph Theory 64 (2010) 219-232.
91
[14] D. Pritikin, Applying a proof of Tverberg to complete bipartite decompositions
of dgraphs and multigraphs, J. Graph Theory 10 (1986),
197-201.
[15] D. Sotteau, Decomposition of Km;n(K
m;n) into cycles (circuits) of length
2k, J. Comb. Theory, Ser. B 30 (1981), 75-81.
[16] D.W. Cranston, Regular bipartite graphs are antimagic, J. Graph Theory
60 (2009), 173-182.
[17] F. Harary, R.W Robinson and N.C. Wormald, The Divisibility theorem
for isomorphic factorization of complete graphs, J.G.T. 1 (1977) 187-188
[18] F. Harary, R.W. Robinson and N.C. Wormald, Isomorphic factorization
III : Complete multipartite graphs, Combinatorial Math., Lecture Notes
in Math.686 (1978) 47-54.
[19] F. R. K. Chung and R. L. Graham, Recent advances in graph decompositions
in combinatorics, Proceeding of 8th British Combinatorial Conference
(University College, Swansea, 1981), edited by H.N.V. Temperley,
it London Math. Soc. Lecuture Notes 1120, Combridge University Press
(1981), 103{124.
92
[20] G. E. Stevens, Properties of Lucas trees, Ars Combin. 92 (2009) 171-192.
[21] G. Kaplan, A. Lev and Y. Roditty, On zero-sum partitions and antimagic
trees, Discrete Math., 309 (2009) 2010-2014.
[22] H. Hanani, Balanced incomplete block designs, Discrete Math. 4 (1975),
255{369.
[23] H. Tverberg, On the decomposition of Kn into complete bipartite
graphs, J. Graph Theory 6 (1982), 493{494.
[24] J. A. Bondy and U. S. R. Murty, it Graph theory with applications,
Macmillan Press, London, 1976.
[25] J. Bosak, Decompositions of Graphs. Kluwer Academic Publishers, Dordrecht.
The Netherlands, 1990.
[26] J. C. Bermond and D. Sotteau, Graph decompositions and G-designs,
Fifth British Combinatorial Conf., Aberdeen Congr. Num. XV (1975),
53{72.
[27] J. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 19
(DS6) (2012) (the Fifteenth edition).
93
[28] J. Knisely, C. Wallis and G. Domke, The partitioned graph isomorphism
problem, Congr. Numer. 89 (1992), 39-44.
[29] K. Ushio, S. Tazawa and S. Yamamoto, On claw decomposition of complete
multipartite graphs, Hiroshima Math. J. 8 (1978), 207{210.
[30] K. Ushio, On balanced claw designs of complete multipartite graphs,
Discrete Math. 38 (1982), 117{119.
[31] K. Ushio, G-designs and related designs, Discrete Math. 116 (1993),
299{311.
[32] L. W. Beineke, Graph decompositions, Congr. Num. 115 (1996), 213{
226.
[33] M. Ba^ca, J. MacDougall, M. Miller, Slamin and W. Wallis, Survey of
certain valuations of graphs, Discussiones Mathematicae Graph Theory,
20 (2000) 219-229
[34] M.J. Lee, C. Lin and W.H. Tsai, On antimagic labeling for power of
cycles. Ars Combin. 98 (2011), 161-165.
[35] M. Jacobson, M. Truszczynski and Z. Tuda, Decompositions of regular
bipartite graphs, Discrete Math. 89 (1991), 17-27.
94
[36] M. Sonntag, Antimagic vertex-labelling of hypergraphs, Discrete Math.
247 (2002) 187- 199.
[37] M. Tarsi, Decomposition of complete multigraphs into stars, Discrete
Math. 26 (1979), 273{278.
[38] M. Tarsi, On the decomposition of a graph into stars, Discrete Math. 36
(1981), 299{304.
[39] M. Tarsi, Decomposition of complete multigraph into simple paths: nonbalanced
handcued designs, J. Combin. Theory, Ser. A 34 (1983), 60{70.
[40] M. Truszczynski, Note on the decomposition of Km;n(K
m;n) into
paths, Discrete Math. 55 (1985), 89-96.
[41] N. Alon, G. Kaplan, A. Lev, Y. Roditty and R. Yuster, Dense graphs
are anti-magic, J. Graph Theory 47 (4) (2004) 297-309.
[42] N. Hartseld and G. Ringel, Pearls in Graph Theory, Academic Press,
Boston, 1990.
[43] N.J. Cavenagh, Decompositions of complete tripartite graphs into k-
cycles, Australasian J. Combin. 18 (1998), 193{200.
95
[44] P.D. Chawathe and V. Krishna, Antimagic labeling of complete m􀀀ary
trees, Number theory and discrete mathematics (Chandigarh, 2000), 77-
80, Trends Math., Birkhuser, Basel, 2002.
[45] R.E. Jamison, G. E. Stevens, Isomorphic factorizations of caterpillars,
Congr. Numer. 158 (2002) 143-151.
[46] R. Sliva, Antimagic labeling graphs with a regular dominating subgraph.
Inform. Process. Lett. 112 (2012), no. 21, 844-847.
[47] S. G. Hartke, P. R. J.  Ostergard, D. Bryant, and S. I. El-Zanati, The
nonexistence of a (K6 􀀀 e)-decomposition of the complete graph K29, J.
Combin. Des. 18 (2010), 94-104.
[48] S. H. Y. Hung and N. S. Mendelsohn, Handcued designs, Deiscrete
Math. 18 (1977), 23{33.
[49] S. Tazawa, Decomposition of a complete multipartite graph into isomorphic
claws, SIAM J. Algebraic Discrete Methods 6 (1985), 413{417.
[50] S.T. Quinn, Isomorphic factorizations of complete equipartite graphs,
J.G.T. 7 (1983) 285-310.
96
[51] S. Yamamoto, H. Ikeda, S. Shige-ede, K. Ushio and N. Hamada, On
claw decomposition of complete graphs and complete bipartite graphs,
Hiroshima Math. J. 5 (1975), 33-42.
[52] T.M. Wang and C. Hsiao, On anti-magic labeling for graph products,
Discrete Math. 308 (2008) 3624-3633.
[53] T. Wang and M.J. Liu, Deming Some classes of disconnected antimagic
graphs and their joins, Wuhan Univ. J. Nat. Sci. 17 (2012), no. 3, 195-199.
[54] T. Wang, M.J. Liu and M.D. Li, A class of antimagic join graphs, Acta
Math. Sin. 29 (2013), no. 5, 1019-1026.
[55] T.M. Wang, Toroidal grids are anti-magic, Lecture Notes in Computer
Science (LNCS) 3595 (2005) 671-679.
[56] T.-W. Shyu, The Decomposition of Complete Graphs, Complete Bipartite
Graphs and Crowns, Ph.D. Thesis, Department of Mathematics,
National Central University, Taiwan (1997).
[57] T.-W. Shyu and C. Lin, Isomorphic path decompositions of crowns, Ars
Combin. 67 (2003), 97-103.
97
[58] T.-W. Shyu, Path decompositions of Kn;n, Ars Combin. 85 (2007),
211-219.
[59] T.-W. Shyu, Decomposition of complete graphs into cycles and stars,
Graphs Combin. 29 (2013), 301-313.
[60] T.-W. Shyu, Decomposition of complete bipartite graphs into paths and
stars with same number of edges, Discrete Math. 313 (2013), 865-871.
[61] Y. Cheng, Lattice grids and prisms are antimagic, math.CO vl 4 Mar
2006.