Directed pathsdecomposition of

complete multidigraph

Zdzisław SkupieńMariusz Meszka

AGH UST Kraków, Poland

For a given graph G of order n, the symbol λG stands for

a λ-multigraph on n vertices, obtained by replacing each edge

of G by λ edges (with the same endvertices).

G 4G

If G Kn then the symbol λKn denotes the complete λ-multigraph

on n vertices.

A decomposition of a multigraph G is a family of edge-disjoint

submultigraphs of Gwhich include all edges of G.

Theorem [M. Tarsi; 1983]Necessary and sufficient conditionsfor the existence of a decomposition of λKn into paths of length m areλn(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

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 most n-3 and, moreover, n is odd or λ is even.

[K. Ng; 1985] improvement on any nonhamiltonian paths in the case n is odd and λ=1

n=9λ=1

01

2

3

4 5

6

7

01

2

3

4 5

6

7

01

2

3

4 5

6

7

01

2

3

4 5

6

7

01

2

3

4 5

6

7

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

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

For a multigraph G, let DG denote a multidigraph obtained from G

by replacing each edge with two opposite arcs connecting

endvertices of the edge.

G DG

For a given graph G of order n, the symbol λG stands for a λ-multigraph

on n vertices, obtained by replacing each edge of G by λ edges (with the same endvertices).

The symbol λDKn denotes the complete λ-multidigraph on n vertices.

digraph D

λ-multidigraphon n vertices, obtained by replacing

each edge of G by λ edges (with the same endvertices).(with the same endvertices).

arc of D arcs

λ-multidigraph

λD

G 4G

DG 4DG

A decomposition of a multigraph G is a family of

edge-disjoint submultigraphs of Gwhich include all edges of G.

A decomposition of a multigraph Gmultidigraph D

which include all edges of G.arc-disjoint submultidigraphs of D

arcs of D

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 into directed hamiltonian paths if and only if neither n=3 and λ is odd nor n=5 and λ=1.

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 to λn(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?

Theorem [Z. Skupień, M. Meszka; 1999]For n 3, the complete multidigraph λDKn 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 multidigraph λDKn 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 and either n=3 and λ is odd or n=5 and λ=1.

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

Conjecture [Z. Skupień, M. Meszka; 2000] The complete multidigraph λDKn is decomposable into directed paths of arbitrarily prescribed lengths 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.