Let n be a positive integer and T be a tree of order 2n. We say that the complete graph K 2n of order 2n has a T -factorization if there are spanning trees T 1 , . . . , T n of K 2n , all isomorphic to T , such that each edge of K 2n belongs to exactly one of T 1 , . . . , T n . Fronček and Kubesa have raised the following question. Suppose that K 2n has a T -factorization. Is it true that T possesses a set X of n vertices such that x∈X deg T (x) = 2n − 1? In this paper, we show that the above

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... uestion has a positive answer if one of the following conditions holds: (i) The degree set D of T has the cardinality at most 3; (ii) The maximum degree ∆ of T is at most 4 or it is at least n − 3.