The intersection problem for K4 - e designs |
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| Institution: | 1. School of Mathematics and Physics, The University of Queensland, Queensland, 4072, Australia;2. Department of Computer Science, University of Oxford, UK;3. ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology (QUT), Australia;4. Department of Mathematics and Statistics, Plymouth University, Plymouth, UK;5. Department of Mathematics, Koç University, Sarıyer, İstanbul 34450, Turkey;1. TUPRAS, Address, Petrol Cd. No:25 D:No:25, Turkey;2. Koc University, Rumelifeneri, Sarıyer Rumeli Feneri Yolu;1. Department of Obstetrics and Gynecology, Division of Gynecologic Oncology, Koc University School of Medicine, Istanbul, Turkey;2. Department of Pathology, Koc University School of Medicine, Istanbul, Turkey;3. Department of Cardiovascular Surgery, Koc University School of Medicine, Istanbul, Turkey;4. Department of Radiation Oncology, Koc University School of Medicine, Istanbul, Turkey;5. Department of Internal Medicine, Koc University School of Medicine, Division of Medical Oncology, Istanbul, Turkey |
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| Abstract: | A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph Kn and B is an edge-disjoint decomposition of Kn into copies of the simple graph G. Following design terminology, we call these copies “blocks”. Here K4 - e denotes the complete graph K4 with one edge removed. It is well-known that a K4 - e design of order n exists if and only if n ≡ 0 or 1 (mod 5), n ⩾ 6. The intersection problem here asks for which k is it possible to find two K4 - e designs (P,B1) and (P,B2) of order n, with |B1 ∩ B2| = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K4 - e designs. |
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