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The spectrum of restricted resolvable designs with $$r=2$$. (English) Zbl 0748.05037
Author’s introduction: A restricted resolvable design $$R_ r RP(p,k)$$ is a resolvable edge-decomposition of the complete graph $$K_ p$$ into $$K_ r$$’s and $$K_{r+1}$$’s in which there are $$k$$ parallel classes. These designs arise quite naturally in the consideration of certain covering problems. An $$R_ 1 RP(p,k)$$ is just a proper $$k$$-edge colouring of $$K_ p$$ and so exists whenever $$k\geq p$$, or $$k=p-1$$ and $$p$$ is even. Necessary and sufficient conditions for the existence of $$R_ 2 RP(p,k)$$, covering all but eighteen values of $$p$$, have been determined in a recent series of papers.
Theorem 1.0. Let $$p>1$$, $$p\not\in\{31,35,37,41,43,47,49,53,55,59,61,65,67,71,79,83,85,89\}$$. There exists an $$R_ 2 RP(p,k)$$ if and only if $$\lfloor{}p/2\rfloor{}\leq k\leq p-1$$ and $$p(k-p+1)\equiv 0$$ modulo 3, with the following exceptions: (i) $$p\equiv 1$$ modulo 6 and $$k=(p-1)/2$$, (ii) $$p$$ is odd and $$k=p-1$$, (iii) $$p\equiv 3$$ modulo 6, $$p\neq 3$$ and $$k=p-2$$, (iv) $$p\equiv 3$$ modulo 6, $$p\neq 9$$ and $$k=p-3$$, and (v) $$(p,k)=(6,3)$$ or (12,6).
In this paper we show that the restriction $$p\neq 49,53,55,61,65,67,71,79,83,85,89$$ can be dropped from the hypothesis of Theorem 1.0. In Section 2 we deal with the orders $$p=79,83,85$$ and 89, and in Section 3 with the orders 49, 53, 55, 61, 65, 67 and 71. Designs for the remaining seven orders, i.e. $$p=31,35,41,43,47$$ and 59, are constructed using direct methods; these constructions can be found in the Technical Report (3). Thus the work here completes the spectrum for restricted resolvable designs with $$r=2$$.
Reviewer: R.N.Mohan (Nuzvid)

##### MSC:
 05B30 Other designs, configurations
##### Keywords:
restricted resolvable designs
Full Text:
##### References:
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