Directed paths decomposition of complete multidigraph Zdzisław Skupień Mariusz 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). G4G If G  K n then the symbol λK n denotes the complete λ-multigraph on n vertices.

A decomposition of a multigraph G is a family of edge-disjoint submultigraphs of G which include all edges of G.

Theorem [M. Tarsi; 1983] Necessary and sufficient conditions for the existence of a decomposition of λK n into paths of length m are λn(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 λK n 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=9 λ=             

Conjecture [M. Tarsi; 1983] The complete multigraph λK n 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 D G denote a multidigraph obtained from G by replacing each edge with two opposite arcs connecting endvertices of the edge. G DGDG

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 λDK n denotes the complete λ-multidigraph on n vertices. digraph D λ-multidigraph on n vertices, obtained by replacing each edge of G by λ edges (with the same endvertices). (with the same endvertices). arc of Darcs λ-multidigraph λDλD

G 4G DGDG4DG4DG

A decomposition of a multigraph G is a family of edge-disjoint submultigraphs of G which include all edges of G. A decomposition of a multigraph G multidigraph 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 λDK n 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 λDK n 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 also sufficient for the existence of a decomposition into given paths?

Theorem [Z. Skupień, M. Meszka; 1999] For n  3, the complete multidigraph λDK n 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 λDK n 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 λDK n into directed paths of the same length m are 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 λDK n is decomposable into directed paths of arbitrarily prescribed lengths 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.