Abstract
A graph G is said to be m-sufficient if m is not exceeding the order of G, each vertex of G is of even degree, and the number of edges in G is a multiple of m. A complete multipartite graph is balanced if each of its partite sets has the same size. In this paper it is proved that the complete multipartite graph G can be decomposed into 4-cycles cyclically if and only if G is balanced and 4-sufficient. Moreover, the problem of finding a maximum cyclic packing of the complete multipartite graph with 4-cycles are also presented.
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References
Alspach B, Gavlas H (2001) Cycle decompositions of K n and K n – I. J Comb Theory Ser B 81:77–99
Billington EJ (1999) Decomposing complete tripartite graphs into cycles of length 3 and 4. Discret Math 197/198:123–135
Billington EJ, Lindner CC (1996) Maximum packing of uniform group divisible triple systems. J Comb Des 4:397–404
Billington EJ, Fu H-L, Rodger CA (2001) Packing complete multipartite graphs with 4-cycles. J Comb Des 9:107–127
Billington EJ, Fu H-L, Rodger CA (2005) Packing λ-fold complete multipartite graphs with 4-cycles. Graphs Comb 21:169–185
Bryant D, Gavlas H, Ling A (2003) Skolem-type difference sets for cycle systems. Electron J Comb 10:1–12
Buratti M (2003) Rotational k-cycle systems of order v<3h; another proof of the existence of odd cycle systems. J Comb Des 11:433–441
Buratti M (2004) Existence of 1-rotational k-cycle systems of the complete graph. Graphs Comb 20:41–46
Buratti M, Del Fra A (2003) Existence of cyclic k-cycle systems of the complete graph. Discret Math 261:113–125
Buratti M, Del Fra A (2004) Cyclic Hamiltonian cycle systems of the complete graph. Discret Math 279:107–119
Cavenagh NJ (1998) Decompositions of complete tripartite graphs into k-cycles. Australas J Comb 18:193–200
Cavenagh NJ (2002) Further decompositions of complete tripartite graphs into 5-cycles. Discret Math 256:55–81
Cavenagh NJ, Billington EJ (2000) Decompositions of complete multipartite graphs into cycles of even length. Graphs Comb 16:49–65
Cavenagh NJ, Billington EJ (2002) On decomposing complete tripartite graphs into 5-cycles. Australas J Comb 22:41–62
Colbourn CJ, Wan P-J (2001) Minimizing drop cost for SONET/WDM networks with 1/8 wavelength requirements. Networks 37:107–116
Fu H-L, Huang W-C (2004) Packing balanced complete multipartite graphs with hexagons. Ars Comb 71:49–64
Fu H-L, Wu S-L (2004) Cyclically decomposing the complete graph into cycles. Discret Math 282:267–273
Hoffman DG, Linder CC, Rodger CA (1989) On the construction of odd cycle systems. J Graph Theory 13:417–426
Kotzig A (1965) Decompositions of a complete graph into 4k-gons. Mat Fyz Casopis Sloven Akad Vied 15:229–233 (in Russian)
Laskar R (1978) Decomposition of some composite graphs into Hamiltonian cycles. In: Hajnal A, Sos VT (eds) Proceedings of the fifth Hungarian coll., Keszthely, 1976. North-Holland, Amsterdam, pp 705–716
Mahmoodian ES, Mirzakhani M (1995) Decomposition of complete tripartite graph into 5-cycles. In: Combinatorics advances. Kluwer Academic, Netherlands, pp 235–241
Mutoh Y, Morihara T, Jimbo M, Fu H-L (2003) The existence of 2×4 grid-block designs and their applications. SIAM J Discret Math 16:173–178
Peltesohn R (1938) Eine Lösung der beiden Heffterschen Differenzenprobleme. Compos Math 6:251–257
Ree DH (1967) Some designs of use in serology. Biometrics 23:779–791
Rosa A (1966a) On cyclic decompositions of the complete graph into (4m+2)-gons. Mat Fyz Casopis Sloven Akad Vied 16:349–352
Rosa A (1966b) On cyclic decompositions of the complete graph into polygons with odd number of edges. Časopis Pĕst Mat 91:53–63 (in Slovak)
Šajna M (2002) Cycle decompositions, III: complete graphs and fixed length cycles. J Comb Des 10:27–78
Sotteau D (1981) Decomposition of K m,n (K * m,n ) into cycles (circuits) of length 2k. J Comb Theory Ser B 30:75–81
Vietri A (2004) Cyclic k-cycle system of order 2km+k; a solution of the last open cases. J Comb Des 12:299–310
Wan P-J (1999) Multichannel optical networks: network theory and application, Kluwer Academic, Dordrecht
Wu S-L, Fu H-L (2006) Cyclic m-cycle systems with m≤32 or m=2q with q a prime power. J Comb Des 14:66–81
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Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.
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Wu, SL., Fu, HL. Maximum cyclic 4-cycle packings of the complete multipartite graph. J Comb Optim 14, 365–382 (2007). https://doi.org/10.1007/s10878-007-9048-6
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DOI: https://doi.org/10.1007/s10878-007-9048-6