Geodesic
Decompositions of nearly complete digraphs into t isomorphic parts
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 3, pp. 563-572.

Voir la notice de l'article provenant de la source Library of Science

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 ^→Pₙ is characterized.
Keywords: decomposition, cyclically 1-generated, remainder, surplus, universal part 
@article{DMGT_2009_29_3_a7,
     author = {Meszka, Mariusz and Skupie\'n, Zdzis{\l}aw},
     title = {Decompositions of nearly complete digraphs into t isomorphic parts},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {563--572},
     publisher = {mathdoc},
     volume = {29},
     number = {3},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2009_29_3_a7/}
}
TY  - JOUR
AU  - Meszka, Mariusz
AU  - Skupień, Zdzisław
TI  - Decompositions of nearly complete digraphs into t isomorphic parts
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2009
SP  - 563
EP  - 572
VL  - 29
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2009_29_3_a7/
LA  - en
ID  - DMGT_2009_29_3_a7
ER  - 
%0 Journal Article
%A Meszka, Mariusz
%A Skupień, Zdzisław
%T Decompositions of nearly complete digraphs into t isomorphic parts
%J Discussiones Mathematicae. Graph Theory
%D 2009
%P 563-572
%V 29
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2009_29_3_a7/
%G en
%F DMGT_2009_29_3_a7
Meszka, Mariusz; Skupień, Zdzisław. Decompositions of nearly complete digraphs into t isomorphic parts. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 3, pp. 563-572. http://geodesic.mathdoc.fr/item/DMGT_2009_29_3_a7/

[1] B. Alspach, H. Gavlas, M. Sajna and H. Verrall, Cycle decopositions IV: complete directed graphs and fixed length directed cycles, J. Combin. Theory (A) 103 (2003) 165-208, doi: 10.1016/S0097-3165(03)00098-0.

[2] C. Berge, Graphs and Hypergraphs (North-Holland, 1973).

[3] J.C. Bermond and V. Faber, Decomposition of the complete directed graph into k-circuits, J. Combin. Theory (B) 21 (1976) 146-155, doi: 10.1016/0095-8956(76)90055-1.

[4] J. Bosák, Decompositions of Graphs (Dordrecht, Kluwer, 1990, [Slovak:] Bratislava, Veda, 1986).

[5] G. Chartrand and L. Lesniak, Graphs and Digraphs (Chapman Hall, 1996).

[6] A. Fortuna and Z. Skupień, On nearly third parts of complete digraphs and complete 2-graphs, manuscript.

[7] F. Harary and R.W. Robinson, Isomorphic factorizations X: Unsolved problems, J. Graph Theory 9 (1985) 67-86, doi: 10.1002/jgt.3190090105.

[8] F. Harary, R.W. Robinson and N.C. Wormald, Isomorphic factorisation V: Directed graphs, Mathematika 25 (1978) 279-285, doi: 10.1112/S0025579300009529.

[9] A. Kedzior and Z. Skupień, Universal sixth parts of a complete graph exist, manuscript.

[10] E. Lucas, Récréations Mathématiques, vol. II (Paris, Gauthier-Villars, 1883).

[11] M. Meszka and Z. Skupień, Self-converse and oriented graphs among the third parts of nearly complete digraphs, Combinatorica 18 (1998) 413-424, doi: 10.1007/PL00009830.

[12] M. Meszka and Z. Skupień, On some third parts of nearly complete digraphs, Discrete Math. 212 (2000) 129-139, doi: 10.1016/S0012-365X(99)00214-9.

[13] M. Meszka and Z. Skupień, Decompositions of a complete multidigraph into nonhamiltonian paths, J. Graph Theory 51 (2006) 82-91, doi: 10.1002/jgt.20122.

[14] M. Plantholt, The chromatic index of graphs with a spanning star, J. Graph Theory 5 (1981) 45-53, doi: 10.1002/jgt.3190050103.

[15] R.C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc. 38 (1963) 99-104, doi: 10.1112/jlms/s1-38.1.99.

[16] Z. Skupień, The complete graph t-packings and t-coverings, Graphs Combin. 9 (1993) 353-363, doi: 10.1007/BF02988322.

[17] Z. Skupień, Clique parts independent of remainders, Discuss. Math. Graph Theory 22 (2002) 361, doi: 10.7151/dmgt.1181.

[18] Z. Skupień, Universal fractional parts of a complete graph, manuscript.