Directed paths decomposition of complete multidigraph

1. Directed paths decomposition ofcomplete multidigraph Zdzisław SkupieńMariusz MeszkaAGH UST Kraków, Poland

2. For a given graph G of order n, the symbol λG standsfor a λ-multigraphon n vertices, obtained byreplacing each edge of G by λ edges (with the same endvertices). G 4G If G Kn then the symbolλKndenotes the complete λ-multigraphonnvertices.

3. Adecomposition of a multigraphGis a family of edge-disjointsubmultigraphs of Gwhich include all edges ofG.

4. Theorem[M. Tarsi; 1983]Necessary and sufficient conditionsfor the existence of a decomposition of λKn into paths of lengthmareλn(n-1)  0 (mod 2m) and n  m+1. [C. Huang][S. Hung, N. Mendelsohn; 1977]handcuffed designs [P. Hell, A. Rosa; 1972]resolvable handcuffed designs

5. Theorem[M. Tarsi; 1983]The complete multigraph λKn is decomposable into undirected paths of any lengths provided that the lengths sum up to λn(n-1)/2, each length is at mostn-3 and, moreover, n is odd or λ is even. [K. Ng; 1985]improvement on any nonhamiltonian paths in the case n is odd andλ=1

6. 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2 7 7 7 7 7      3 3 3 3 3 6 6 6 6 6 4 4 4 4 4 5 5 5 5 5 n=9λ=1  0 1 7 2 6 3 5 4   1 2 0 3 7 4 6 5   2 3 1 4 0 5 7 6   3 4 2 5 1 6 0 7 

7. Conjecture[M. Tarsi; 1983]The complete multigraph λKn is decomposable into undirected paths of arbitrarily prescribed lengthsprovided that the lengths sum up to λn(n-1)/2.

8. For a multigraph G, letDG denote a multidigraph obtained from Gby replacing each edge with two opposite arcs connecting endvertices of theedge. G DG

9. For a given graph G of order n, the symbol λGstandsfor a λ-multigraphon n vertices, obtained byreplacing each edge of G by λ edges (with the same endvertices). digraph D λD λ-multidigraph λ-multidigraph on n vertices, obtained byreplacingeach edge of G by λ edges (with the same endvertices). arc of D arcs (with the same endvertices). The symbolλDKndenotes the complete λ-multidigraph onnvertices.

10. DG 4DG G 4G

11. Adecomposition of a multigraphGis a family of edge-disjointsubmultigraphs of Gwhich include all edges ofG. Adecomposition of a multigraphG multidigraphD arc-disjoint submultidigraphs of D which include all edges ofG. arcs of D

12. Problem[E. Strauss; ~1960]Can the complete digraph on n vertices be decomposed into n directed hamiltonian paths? [J-C. Bermond, V. Faber; 1976]even n [T. Tillson; 1980] odd n, n  7 Theorem[J. Bosák; 1986]The multigraph λDKn is decomposable intodirected hamiltonian paths if and only if neither n=3 and λ isodd nor n=5 and λ=1.

13. Problem[Z. Skupień, M. Meszka; 1997]If the complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths then the lengths must sum up toλn(n-1), and moreover all paths cannot be hamiltonian if either n=3 and λ is odd or n=5 and λ=1.Are the above necessary conditions alsosufficient for the existence of a decomposition into given paths?

14. Theorem[Z. Skupień, M. Meszka; 1999]For n 3, the complete multidigraphλDKn is decomposable into directed nonhamiltonian paths of arbitrarily prescribed lengths ( n-2) provided that the lengths sum up to λn(n-1). Theorem[Z. Skupień, M. Meszka; 2004]For n 4, the complete multidigraphλDKn is decomposable into directed paths of arbitrarily prescribed lengths except the length n-2,provided that the lengths sum up to λn(n-1), unless all paths are hamiltonian andeither n=3 and λ is odd or n=5 and λ=1.

15. Corollary[Z. Skupień, M. Meszka; 2004]Necessary and sufficient conditions for the existence of a decompositionof λDKninto directed paths of the same length mare λn(n-1)  0 (mod m) and mn-1,unless m=n-1and either n=3 and λ is odd or n=5 andλ=1.

16. Conjecture[Z. Skupień, M. Meszka; 2000]The complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths provided thatthe lengths sum up toλn(n-1), unless all paths are hamiltonian andeither n=3 and λ is odd or n=5 and λ=1.