Abstract
Let G of order n be the vertex-disjoint union of an even and an odd cycle. It is known that there exists a G-decomposition of \(K_v\) for all \(v \equiv 1 \pmod {2n}\). We use an extension of the Bose construction for Steiner triple systems and a recent result on the Oberwolfach Problem for 2-regular graphs with two components to show that there exists a G-decomposition of \(K_{v}\) for all \(v \equiv n \pmod {2n}\), unless \(G = C_4\cup C_5\) and \(v = 9\).
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References
Abrham, J., Kotzig, A.: Graceful valuations of \(2\)-regular graphs with two components. Discrete Math. 150, 3–15 (1996)
Adams, P., Bryant, D., Buchanan, M.: A survey on the existence of \(G\)-designs. J. Combin. Des. 16, 373–410 (2008)
Adams, P., Bryant, D., Gavlas, H.: Decompositions of the complete graph into small 2-regular graphs. J. Combin. Math. Combin. Comput. 43, 135–146 (2002)
Alspach, B., Gavlas, H.: Cycle decompositions of \(K_n\) and of \(K_n-I\). J. Combin. Theory Ser. B 81, 77–99 (2001)
Blinco, A., El-Zanati, I.: A note on the cyclic decomposition of complete graphs into bipartite graphs. Bull. Inst. Combin. Appl. 40, 77–82 (2004)
Blinco, A., El-Zanati, S., Vanden Eynden, C.: On the decomposition of complete graphs into almost-bipartite graphs. Discrete Math. 284, 71–81 (2004)
Bose, R.C.: On the construction of balanced incomplete block designs. Ann. Eugenics 9, 353–399 (1939)
Bryant, D., El-Zanati, S.: Graph decompositions. In: Colbourn, C.J., Dinitz, J.H.(eds.), Handbook of Combinatorial Designs. 2nd edn, pp. 477–485, Chapman & Hall/CRC, Boca Raton (2007)
Bunge, R.C., Chantasartrassmee, A., El-Zanati, S., Vanden Eynden, C.: On cyclic decompositions of complete graphs into tripartite graphs. J. Graph Theory 72, 90–111 (2013)
Buratti, M., Gionfriddo, L.: Strong difference families over arbitrary graphs. J. Combin. Des. 16, 443–461 (2008)
El-Zanati, S., Vanden Eynden, C., Punnim, N.: On the cyclic decomposition of complete graphs into bipartite graphs. Australas. J. Combin. 24, 209–219 (2001)
Gannon, D.I., El-Zanati, S.: All 2-regular graphs with uniform odd components admit \(\rho \)-labelings. Australas. J. Combin. 53, 207–219 (2012)
Hoffman, D.G., Lindner, C.C., Rodger, C.A.: On the construction of odd cycle systems. J. Graph Theory 13, 417–426 (1989)
Jongthawonwuth, U., El-Zanati, S., Uiyyasathian, C.: On extending the Bose construction for triple systems to decompositions of complete multipartite graphs into 2-regular graphs of odd order. Australas. J. Combin. 59, 378–390 (2014)
Lindner, C.C., Rodger, C.A.: Design Theory: Discrete Mathematics and its Applications, 2nd edn. CRC Press, Boca Raton, FL (2009)
Lindner, C.C., Rodger, C.A., Stinson, C.R.: Nesting of cycle systems of odd length. Discrete Math. 77, 191–203 (1989)
Šajna, M.: Cycle decompositions III: complete graphs and fixed length cycles. J. Combin. Des. 10, 27–78 (2002)
Traetta, A.: Complete solution to the two-table Oberwolfach problems. J. Combin. Theory Ser. A 120, 984–997 (2013)
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El-Zanati, S.I., Jongthawonwuth, U., Jordon, H., Eynden, C.V. (2015). On Decomposing the Complete Graph into the Union of Two Disjoint Cycles. In: Jan, K., Miller, M., Froncek, D. (eds) Combinatorial Algorithms. IWOCA 2014. Lecture Notes in Computer Science(), vol 8986. Springer, Cham. https://doi.org/10.1007/978-3-319-19315-1_14
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DOI: https://doi.org/10.1007/978-3-319-19315-1_14
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