Abstract
Let G be a 2k-regular graph in which every cycle of length at most k is an induced cycle. In this paper, we prove that G can be decomposed into paths of length k, and moreover, every vertex is a terminal of exactly two paths.
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References
Botler, F., Talon, A.: Decomposing 8-regular graphs into paths of length 4. Discret. Math. 340, 2275–2285 (2017)
El-Zanati, S.I., Ermete, M., Hasty, J., Plantholt, M.J., Tipnis, S.: On decomposing regular graphs into isomorphic double-stars. Discuss. Math. Graph Theory 35, 73–79 (2015)
El-Zanati, S.I., Kopp, M., Plantholt, M.J., Rice, S.: On decomposing regular graphs into star forests. Int. J. Math. Comput. Sci. 2, 249–256 (2016)
Häggkvist, R.: Decompositions of complete bipartite graphs. In: Surveys in Combinatorics, 1989 (Norwich, 1989). In: London Mathematics Society Lecture Note Series, vol. 141, pp. 115–147. Cambridge University Press, Cambridge (1989)
Jao, K.F., Kostochka, A.V., West, D.B.: Decomposition of Cartesian products of regular graphs into isomorphic trees. J. Combin. 4, 469–490 (2013)
Kouider, M., Lonc, Z.: Path decompositions and perfect path double covers. Australas. J. Combin. 19, 261–274 (1999)
Petersen, J.: Die Theorie der regulären graphs. Acta Math. 15, 193–220 (1891)
Snevily, H.: Combinatorics of finite sets, PhD Thesis, University of Illinois, (1991) MR2687109
Xie, M., Zhou, C.: Decomposing 10-regular graphs into paths of length 5. Discuss. Math. Graph Theory 42, 1089–1097 (2022)
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The Funding has been recevied from National Natural Science Foundation of China with Grant nos. 11971110.
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Xie, M., Zhou, C. Decomposing 2k-Regular Graphs into Paths of Length k. Graphs and Combinatorics 38, 139 (2022). https://doi.org/10.1007/s00373-022-02546-3
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DOI: https://doi.org/10.1007/s00373-022-02546-3