Cycle factorizations of cycle products

AbstractLet n and k1,k2,…,kn be integers with n > 1 and ki ⩾ 2 for 1 ⩽ i ⩽ n. We show that there exists a Cs-factorization of Πi=1n C2ki if and only if s = 2t with 2 ⩽ t ⩽ k1 + ··· + kn. We also settle the problem of cycle factorizations of the d-cube

Open conversion in laparoscopic cholecystectomy and bile duct exploration: subspecialisation safely reduces the conversion rates

Background: Open conversion rates during laparoscopic cholecystectomy vary depending on many factors. Surgeon experience and operative difficulty influence the decision to convert on the grounds of patient safety but occasionally due to technical factors. We aim to evaluate the difficulties leading to conversion, the strategies used to minimise this event and how subspecialisation influenced conversion rates over time. Methods: Prospectively collected data from 5738 laparoscopic cholecystectomies performed by a single surgeon over 28 years was analysed. Routine intraoperative cholangiography and common bile duct exploration when indicated are utilised. Patients undergoing conversion, fundus first dissection or subtotal cholecystectomy were identified and the causes and outcomes compared to those in the literature. Results: 28 patients underwent conversion to open cholecystectomy (0.49%). Morbidity was relatively high (33%). 16 of the 28 patients (57%) had undergone bile duct exploration. The most common causes of conversion in our series were dense adhesions (9/28, 32%) and impacted bile duct stones (7/28, 25%). 173 patients underwent fundus first cholecystectomy (FFC) (3%) and 6 subtotal cholecystectomy (0.1%). Morbidity was 17.3% for the FFC and no complications were encountered in the subtotal cholecystectomy patients. These salvage techniques have reduced our conversion rate from a potential 3.5% to 0.49%. Conclusion: Although open conversion should not be seen as a failure, it carries a high morbidity and should only be performed when other strategies have failed. Subspecialisation and a high emergency case volume together with FFC and subtotal cholecystectomy as salvage strategies can reduce conversion and its morbidity in difficult cholecystectomies

Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.Comment: 30 pages, 3 table

Partial spreads and vector space partitions

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.Comment: 30 pages, 1 tabl

d-cube decompositions of K-n/K-m

Necessary conditions on n, m and d are given for the existence of an edge-disjoint decomposition of K-n\K-m into copies of the graph of a d-dimensional cube. Sufficiency is shown when d = 3 and, in some cases, when d = 2(t). We settle the problem of embedding 3-cube decompositions of K-m into 3-cube decompositions of K-n; where n greater than or equal to m

Packing and covering the complete graph with cubes

A decomposition of K \L, the complete graph of order n with a subgraph L (called the leave) removed, into edge disjoint copies of a graph G is called a maximum packing of K with G if L contains as few edges as possible. A decomposition of K ∪ P, the complete graph union a graph P (called the padding), into edge disjoint copies of a graph G is called a minimum covering of K with G if P contains as few edges as possible. We construct maximum packings and minimum coverings of K with the 3-cube for all n

Factorizations of and by powers of complete graphs

Let K-k(d) denote the Cartesian product of d copies of the complete graph K-k. We prove necessary and sufficient conditions for the existence of a K-k(r)-factorization of K-pn(s), where p is prime and k > 1, n, r and s are positive integers. (C) 2002 Elsevier Science B.V. All rights reserved

Labelings of unions of up to four uniform cycles

We show that every 2-regular graph consisting of at most four uniform components has a ρ-labeling (or a more restricted labeling). This has an application in the cyclic decomposition of certain complete graphs into the disjoint unions of cycles