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ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 29(3) (2009) 563-572


Mariusz Meszka  and  Zdzisław Skupień

Faculty of Applied Mathematics
AGH University of Science and Technology
al. Mickiewicza 30, 30-059 Kraków, Poland


An arc decomposition of the complete digraph 𝒟Kn 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 (𝒟Kn-R)/t and the ceiling tth class (𝒟Kn+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 𝒟Kn into cycles Cn−1 and into paths Pn is characterized.

Keywords: decomposition, cyclically 1-generated, remainder, surplus, universal part.

2000 Mathematics Subject Classification: 05C70, 05C20.


[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.

Received 8 May 2008
Revised 22 September 2008
Accepted 13 October 2008