Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.

Some equitably 3-colourable cycle decompositions

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has n(i) vertices coloured c(i), i = 1, 2,..., k, and (i) - n(j) less than or equal to 1 for any i, j is an element of {1, 2,..., k}, then C is equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in C is equitably k-coloured. For m = 4,5 and 6, we completely settle the existence problem for equitably 3-colourable m-cycle decompositions of complete graphs and complete graphs with the edges of a 1-factor removed. (C) 2004 Elsevier B.V. All rights reserved.

Some equitably 3-colourable cycle decompositions of complete equipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),.. , c(k). If an m-cycle C in G has n(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar n(i) - n(j)vertical bar <= 1 for any i, j is an element of {1, 2,..., k}, then C is said to be equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in W is equitably k-coloured. For m = 3, 4 and 5 we completely settle the existence question for equitably 3-colourable m-cycle decompositions of complete equipartite graphs. (c) 2005 Elsevier B.V. All rights reserved.

Some equitably 2-colourable cycle decompositions of multipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has x(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar x(i) - x(j)vertical bar

Edge-coloured cube decompositions

An edge-colored graph is a graph H together with a function f:E(H) → C where C is a set of colors. Given an edge-colored graph H, the graph induced by the edges of color c C is denoted by H(c). Let G, H, and J be graphs and let μ be a positive integer. A (J, H, G, μ) edge-colored graph decomposition is a set S = {H 1,H 2,...,H t} of edge-colored graphs with color set C = {c 1, c 2,..., c k} such that Hi ≅ H for 1 ≤ i ≤ t; Hi (cj) ≅ G for 1 ≤ i ≤ t and ≤ j ≤ k; and for j = 1, 2,..., k, each edge of J occurs in exactly μ of the graphs H 1(c j ), H 2(c j ),..., H t (c j ). Let Q 3 denote the 3-dimensional cube. In this paper, we find necessary and sufficient conditions on n, μ and G for the existence of a (K n ,Q 3,G, μ) edge-colored graph decomposition. © Birkhäuser Verlag, Basel 2007.

Avaliação comparativa da remodelação óssea periimplantar em implantes com junção cone-morse e hexágono externo: estudo clínico randomizado controlado, duplo cego, boca dividida

Background: Several theories, such as the biological width formation, the inflammatory reactions due to the implant-abutment microgap contamination, and the periimplant stress/strain concentration causing bone microdamage accumulation, have been suggested to explain early periimplant bone loss. However, it is yet not well understood to which extent the implant-abutment connection type may influence the remodeling process around dental implants. Aim: to evaluate clinical, bacteriological, and biomechanical parameters related to periimplant bone loss at the crestal region, comparing external hexagon (EH) and Morse-taper (MT) connections. Materials and methods: Twelve patients with totally edentulous mandibles received four custom made Ø 3.8 x 13 mm implants in the interforaminal region of the mandible, with the same design, but different prosthetic connections (two of them EH or MT, randomly placed based on a split-mouth design), and a immediate implant- supported prosthesis. Clinical parameters (periimplant probing pocket depth, modified gingival index and mucosal thickness) were evaluated at 6 sites around the implants, at a 12 month follow-up. The distance from the top of the implant to the first bone-to-implant contact – IT-FBIC was evaluated on standardized digital peri-apical radiographs acquired at 1, 3, 6 and 12 months follow-up. Samples of the subgingival microbiota were collected 1, 3 and 6 months after implant loading. DNA were extracted and used for the quantification of Tanerella forsythia, Porphyromonas gingivalis, Aggragatibacter actinomycetemcomitans, Prevotella intermedia and Fusobacterium nucleatum. Comparison among multiple periods of observation were performed using repeated-measures Analysis of Variance (ANOVA), followed by a Tukey post-hoc test, while two-period based comparisons were made using paired t- test. Further, 36 computer-tomographic based finite element (FE) models were accomplished, simulating each patient in 3 loading conditions. The results for the peak EQV strain in periimplant bone were interpreted by means of a general linear model (ANOVA). Results: The variation in periimplant bone loss assessed by means of radiographs was significantly different between the connection types (P<0.001). Mean IT-FBIC was 1.17±0.44 mm for EH, and 0.17±0.54 mm for MT, considering all evaluated time periods. All clinical parameters presented not significant differences. No significant microbiological differences could be observed between both connection types. Most of the collected samples had very few pathogens, meaning that these regions were healthy from a microbiological point of view. In FE analysis, a significantly higher peak of EQV strain (P=0.005) was found for EH (mean 3438.65 µ∑) compared to MT (mean 840.98 µ∑) connection. Conclusions: Varying implant-abutment connection type will result in diverse periimplant bone remodeling, regardless of clinical and microbiological conditions. This fact is more likely attributed to the singular loading transmission through different implant-abutment connections to the periimplant bone. The present findings suggest that Morse-taper connection is more efficient to prevent periimplant bone loss, compared to an external hexagon connection.

Aspects of Motives: Finite-dimensionality, Chow-Kunneth Decompositions and Intersections of Cycles

This thesis analyzes the Chow motives of 3 types of smooth projective varieties: the desingularized elliptic self fiber product, the Fano surface of lines on a cubic threefold and an ample hypersurface of an Abelian variety. For the desingularized elliptic self fiber product, we use an isotypic decomposition of the motive to deduce the Murre conjectures. We also prove a result about the intersection product. For the Fano surface of lines, we prove the finite-dimensionality of the Chow motive. Finally, we prove that an ample hypersurface on an Abelian variety possesses a Chow-Kunneth decomposition for which a motivic version of the Lefschetz hyperplane theorem holds.