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| Paper IPM / M / 16477 |
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| Abstract: | |||||
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In 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing
a well-known conjecture by Alspach, obtained the necessary
and sufficient conditions for the decomposition of
the complete multigraph �??ð¾ï¿½?? �?? ð¼ into cycles of arbitrary
lengths, where ð¼ is empty, when �??(�?? �?? 1) is even and ð¼ is a
perfect matching, when �??(�?? �?? 1) is odd. Moreover, Bryant
in 2010, verifying a conjecture by Tarsi, proved that the
obvious necessary conditions for packing pairwise edgedisjoint
paths of arbitrary lengths in �??ð¾ï¿½?? are also sufficient.
In this article, first, we obtain the necessary and sufficient
conditions for packing edge-disjoint cycles of arbitrary
lengths in �??ð¾ï¿½?? �?? ð¼. Then, applying this result, we
investigate the analogous problem of the decomposition of
the complete uniform multihypergraph �??ð¾(�??)�?? into Berge
cycles and paths of arbitrary given lengths. In particular, we
show that for every integer �?? �?� 1, �?? �?� 108 and 3 �?� �?? < �??,
�??ð¾(�??)�?? can be decomposed into Berge cycles and paths of
arbitrary lengths, provided that the obvious necessary conditions
hold, thereby generalizing a result by Kühn and
Osthus on the decomposition of ð¾(�??)�?? into Hamilton Berge
cycles.
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