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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
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References:
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