On the construction of odd cycle systems.(English)Zbl 0704.05031

Let $$K_ n$$ be a complete nonoriented graph. A cycle system of order n is defined as a decomposition of the edge set of $$K_ n$$ into disjoint cycles of the length k. The necessary conditions for $$K_ n$$ to be decomposable into k-cycles are $$n\geq k,$$ n is odd, $$2k| n(n-1).$$ (*)
In the paper there is investigated the case of odd k. It is shown that necessary conditions (*) are also sufficient if and only if each admissible graph $$K_ n$$ (satisfying (*) for $$k\leq n<3k)$$ is decomposable into k-cycles. The cases of decompositions into 15-cycles and 21-cycles are solved completely.
Reviewer: L.Niepel

MSC:

 05C38 Paths and cycles 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)

Keywords:

complete graph; k-cycles; decompositions
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References:

 [1] Alspach, Ann. Discrete Math. 9 pp 155– (1980) [2] Bermond, Congressus Numerantium 15 pp 53– (1975) [3] Bermond, Ars. Combinat. 5 pp 293– (1978) [4] and , Cycle and circuit designs: Odd Case. Beitrage zur Graphentheorie und deren Anwendungen. Proc. Colloq. Oberhof Illmenau (1978) 11–32. [5] and , Graph Theory with Applications, North Holland, (1976). · Zbl 1226.05083 [6] The intersection problem for pentagon systems. Ph.D. thesis, Auburn University (1987). [7] Häggkvist, Ann. Discrete Math. 27 pp 227– (1985) [8] Kotzig, Mat.- Fyz. Cas 15 pp 227– (1965) [9] Rosa, Mat.-Fyz. Cas 16 pp 349– (1966) [10] Rosa, Casopis Pest. Math. 91 pp 53– (1966) [11] Rosa, Discrete Math. 12 pp 269– (1975) [12] Décompositions de graphes et hypergraphes. Thèses de l’Université Paris-Sud (1980). [13] Sotteau, J. Combinat. Theory B 29 pp 75– (1981) [14] Stern, Boll. Un. Math. Ital. A(5) 17 pp 109– (1980) [15] Vizing, Discrete Analiz. 3 pp 25– (1964) [16] Wilson, Math. Centre Tracts 55 pp 18– (1974) [17] Wilson, Congresses Numerantium 15 pp 647– (1975) [18] Wilson, J. Combinat. Theory A 13 pp 246– (1972)
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