Abstract
In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K v . This supplementary solution builds on a result of Raines and Staniszlo.
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Andersen L.D., Hilton A.J.W., Mendelsohn E.: Embedding partial Steiner triple systems. Proc. Lond. Math. Soc. 41, 557–576 (1980)
Bermond J.-C., Huang C., Sotteau D.: Balanced cycle and circuit designs: even cases. Ars Combinatoria 5, 292–318 (1978)
Billington E.J., Fu H., Rodger C.A.: Packing complete multipartite graphs with 4-cycles. J Comb. Des. 9, 107–127 (2001)
Bryant D., Horsley D.: A proof of Lindner’s conjecture on embeddings of partial Steiner triple systems. J. Comb. Des. 17(1), 63–89 (2009)
Bryant D.E., Rodger C.A.: On the Doyen–Wilson theorem for m-cycle systems. J. Comb. Des. 4, 253–271 (1994)
Bryant D.E., Rodger C.A.: The Doyen–Wilson theorem extended to 5-cycle. J. Comb. Theory A 68, 218–225 (1994)
Bryant D., Hoffman D.G., Rodger C.A.: 5-Cycle systems with holes full. Des. Codes Cryptogr. 8, 103–108 (1996)
Bryant D.E., Rodger C.A., Spicer E.R.: Embeddings of m-cycle systems and incomplete m-cycle systems: m ≤ 14. Discrete Math. 171, 55–75 (1997)
Bryant D., Horsley D., Maenhaut B.: Decompositions into 2-regular subgraphs and equitable partial cycle decompositions. J. Comb. Theory Ser. B 93, 67–72 (2005)
Colbourn C.J., Hamm R.C., Rosa A.: Embedding, immersing, and enclosing. In: Proceedings of the Sixteenth Southeastern international Conference on Combinatorics, Graph Theory and Computing Boca Raton, FL, 1985). Congr. Num. 47, 229–236 (1985).
Doyen J., Wilson R.M.: Embeddings of Steiner triple systems. Discrete Math. 5, 229–239 (1973)
Fu H.L., Rodger C.A.: 4-Cycle Group-divisible designs with two associate classes. Comb. Prob. Comput. 10, 317–343 (2001)
Horton J.D., Lindner C.C., Rodger C.A.: A small embedding for partial 4-cycle systems. JCMCC 5, 23–26 (1989)
Horak P., Lindner C.C.: A small embedding for partial even-cycle systems. J. Comb. Des. 7, 205–215 (1999)
Hurd S.P., Sarvate D.G.: Minimal enclosings of triple systems II: increasing the index by 1. Ars. Comb. 68, 263–282 (2003)
Hurd S.P., Munson P., Sarvate D.G.: Minimal enclosings of triple systems I: adding one point. Ars. Comb. 68, 145–159 (2003)
Lindner C.C.: A Partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6n + 3. J. Comb. Theory Ser. A 18(3), 349–351 (1975)
Lindner C.C., Rodger C.A. (1993). A partial m = (2k + 1)-cycle system of order n can be embedded in an m-cycle system of order (2n + 1)m. Discrete Math. 117, 151–159
Lindner C.C., Rodger C.A.: Design Theory. CRC Press, Boca Raton (1997)
Newman N.A., Rodger C.A.: Enclosings of λ-fold triple systems. Utilitas Math. (to be published).
Raines M.E., Staniiszlo Z.: Equitable partial cycle systems. Aust. J. Comb. 19, 149–156 (1999)
Schonheim J., Bialistocki A.: Packing and covering the complete graph with 4-cycles. Can. Math. Bull. 18, 703–708 (1975)
Sotteau D.: Decompositions of \({K_{m,n}(K_{m,n}^{*})}\) into cycles of length 2k. J. Comb. Theory Ser. B 30, 75–81 (1981)
Treash C.: The completion of finite incomplete Steiner triple systems with applications to loop theory. J. Comb. Theory Ser. A 10, 259–265 (1971)
West D.B.: Introduction to Graph Theory. Prentice Hall (2001).
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Communicated by Ron Mullin, Rainer Steinwandt.
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Newman, N.A., Rodger, C.A. Enclosings of λ-fold 4-cycle systems. Des. Codes Cryptogr. 55, 297–310 (2010). https://doi.org/10.1007/s10623-009-9353-4
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DOI: https://doi.org/10.1007/s10623-009-9353-4