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Paper IPM / M / 16477  


Abstract:  
In 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing
a wellknown conjecture by Alspach, obtained the necessary
and sufficient conditions for the decomposition of
the complete multigraph ï¿½??ð¾ï¿½?? ï¿½?? ð¼ into cycles of arbitrary
lengths, where ð¼ is empty, when ï¿½??(ï¿½?? ï¿½?? 1) is even and ð¼ is a
perfect matching, when ï¿½??(ï¿½?? ï¿½?? 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 ï¿½??ð¾ï¿½?? are also sufficient.
In this article, first, we obtain the necessary and sufficient
conditions for packing edgedisjoint cycles of arbitrary
lengths in ï¿½??ð¾ï¿½?? ï¿½?? ð¼. Then, applying this result, we
investigate the analogous problem of the decomposition of
the complete uniform multihypergraph ï¿½??ð¾(ï¿½??)ï¿½?? into Berge
cycles and paths of arbitrary given lengths. In particular, we
show that for every integer ï¿½?? ï¿½?ï¿½ 1, ï¿½?? ï¿½?ï¿½ 108 and 3 ï¿½?ï¿½ ï¿½?? < ï¿½??,
ï¿½??ð¾(ï¿½??)ï¿½?? can be decomposed into Berge cycles and paths of
arbitrary lengths, provided that the obvious necessary conditions
hold, thereby generalizing a result by KÃ¼hn and
Osthus on the decomposition of ð¾(ï¿½??)ï¿½?? into Hamilton Berge
cycles.
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