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Discussiones Mathematicae 343 Graph Theory 26 (2006 ) 343–349

9 8

13th WORKSHOP

‘3in1’ GRAPHS 2004

Krynica, November 11-13, 2004

PROBLEM PRESENTED AT THE WORKSHOP IN KRYNICA 2004

This is a problem by Michael Kubesa, Technical University Ostrava, presented by Dalibor Froncek.

Let K 2n be a complete graph and T a tree, both with 2n vertices. A T -factorization of K 2n is a collection of edge disjoint spanning subgraphs (i.e., factors) T 1 , T 2 , . . . , T n of K 2n , all isomorphic to T . Every edge of K 2n then appears in exactly one copy of T .

M. Kubesa asked the following question: Suppose that there exists a T -factorization of K 2n . Is it then true that the vertex set of T can be decomposed into two subsets, X and Y , such that

(1) |X| = |Y | = n, (2) P

x∈X deg(x) = P

y∈Y deg(y) ?

Notice that the sets X, Y in general are not the partite sets of the bipartition

of T .

Cytaty

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