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Solution to the problem of Kubesa

Mariusz Meszka — 2008

Discussiones Mathematicae Graph Theory

An infinite family of T-factorizations of complete graphs K 2 n , where 2n = 56k and k is a positive integer, in which the set of vertices of T can be split into two subsets of the same cardinality such that degree sums of vertices in both subsets are not equal, is presented. The existence of such T-factorizations provides a negative answer to the problem posed by Kubesa.

Decompositions of nearly complete digraphs into t isomorphic parts

Mariusz MeszkaZdzisław Skupień — 2009

Discussiones Mathematicae Graph Theory

An arc decomposition of the complete digraph Kₙ into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class ( Kₙ-R)/t and the ceiling tth class ( Kₙ+S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest possible. The existence of cyclically 1-generated decompositions of Kₙ into cycles C n - 1 and into paths P is characterized....

Decompositions of a complete multidigraph into almost arbitrary paths

Mariusz MeszkaZdzisław Skupień — 2012

Discussiones Mathematicae Graph Theory

For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n - 1 and different from n - 2, unless n = 5, λ = 1, and all lengths are to be n - 1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n - 2 can also be prescribed.

Decompositions of multigraphs into parts with two edges

Jaroslav IvančoMariusz MeszkaZdzisław Skupień — 2002

Discussiones Mathematicae Graph Theory

Given a family 𝓕 of multigraphs without isolated vertices, a multigraph M is called 𝓕-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of 𝓕. We present necessary and sufficient conditions for the existence of such decompositions if 𝓕 comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.

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