Sandwiching saturation number of fullerene graphs

The saturation number of a graph $G$ is the cardinality of any smallest maximal matching of $G$, and it is denoted by $s(G)$. Fullerene graphs are cubic planar graphs with exactly twelve 5-faces; all the other faces are hexagons. They are used to capture the structure of carbon molecules. Here we show that the saturation number of fullerenes on $n$ vertices is essentially $n/3$

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Diâmetro de grafos fulerenes e transversalidade de ciclos ímpares de fuleróides-(3, 4, 5, 6)

Fullerene graphs are mathematical models for molecules composed exclusively of carbon atoms, discovered experimentally in the early 1980s by Kroto, Heath, O’Brien, Curl and Smalley. Many parameters associated to these graphs have been discussed, trying to describe the stability of the fullerene’s molecule. Formally, fulerene graphs are 3-connected, cubic, planar graphs with pentagonal and hexagonal faces. Andova and Škrekovski Conjecture [1] states that the diameter of all fullerene graph, on n vertices, is at least equal to jq 5n 3 k −1. This conjecture became relevant, since Andova and Škrekovski conceived it from the study of highly regular, spherical and symmetrical fullerene graphs. We introduce the concepts of combinatorial curvature of vertex and combinatorial curvature of face of a planar graph and then we define a specific class of fullerene graphs, called fullerene nanodiscs. We have shown that the Andova and Škrekovski Conjecture is not valid for any fullerene nanodisc with more than 300 vertices. However, we exhibit infinite classes of fullerene graphs, similar to the graphs studied by Andova and Škrekovski, which satisfy this conjecture. Adding to fullerene graphs, triangular and quadrangular faces we conceive fuleroid-(3, 4, 5, 6) graphs. We studied the bipartite edge frustration and the maximum independent set problems on the fuleroid-(3, 4, 5, 6) graphs, obtaining tight limits for both problems.Os grafos fulerenes são modelos matemáticos para moléculas compostas exclusivamente por átomos de carbono, descobertas experimentalmente no início da década de 80 por Kroto, Heath, O’Brien, Curl e Smalley. Muitos parâmetros associados a estes grafos vêm sendo discutidos, buscando descrever a estabilidade das moléculas de fulerene. Precisamente falando, grafos fulerenes são grafos cúbicos, planares, 3-conexos cujas faces são pentagonais e hexagonais. A Conjectura de Andova e Škrekovski [1] afirma que o diâmetro de todo grafo fulerene, contendo n vértices, é pelo menos igual a jq 5n 3 k − 1. Esta conjectura tornou-se relevante, pois Andova e Škrekovski conceberam-na a partir do estudo de grafos fulerenes altamente regulares, esféricos e simétricos. Introduzimos os conceitos de curvatura combinatória de vértice e curvatura combinatória de face de um grafo planar. Definimos, então, uma classe particular de grafos fulerenes, chamada de nanodiscos de fulerene. Mostramos que a Conjectura de Andova e Škrekovski não é válida para nenhum nanodisco de fulerene com mais de 300 vértices. No entanto, exibimos infinitas classes de grafos fulerenes, semelhantes aos grafos estudados por Andova e Škrekovski, que satisfazem a referida conjectura. Acrescentando, aos grafos fulerenes, faces triangulares e quadrangulares concebemos os grafos fuleróides-(3, 4, 5, 6). Estudamos os problemas da frustração bipartida de arestas e do conjunto independente máximo sobre os grafos fuleróides-(3, 4, 5, 6), obtendo limites apertados para ambos os problemas

ODD CYCLE TRANSVERSALS AND INDEPENDENT SETS IN FULLERENE GRAPHS

Abstract. A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that every fullerene graph on n vertices can be made bipartite by deleting at most √ 12n/5 edges, and has an independent set with at least n/2 − √ 3n/5 vertices. Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Doˇslić and Vukičević, and of Daugherty. We deduce two further conjectures on the independence number of fullerene graphs, as well as a new upper bound on the smallest eigenvalue of a fullerene graph. 1

The benefits of an additional practice in descriptive geomerty course: non obligatory workshop at the Faculty of Civil Engineering in Belgrade

At the Faculty of Civil Engineering in Belgrade, in the Descriptive geometry (DG) course, non-obligatory workshops named “facultative task” are held for the three generations of freshman students with the aim to give students the opportunity to get higher final grade on the exam. The content of this workshop was a creative task, performed by a group of three students, offering free choice of a topic, i.e. the geometric structure associated with some real or imagery architectural/art-work object. After the workshops a questionnaire (composed by the professors at the course) is given to the students, in order to get their response on teaching/learning materials for the DG course and the workshop. During the workshop students performed one of the common tests for testing spatial abilities, named “paper folding". Based on the results of the questionnairethe investigation of the linkages between:students’ final achievements and spatial abilities, as well as students’ expectations of their performance on the exam, and how the students’ capacity to correctly estimate their grades were associated with expected and final grades, is provided. The goal was to give an evidence that a creative work, performed by a small group of students and self-assessment of their performances are a good way of helping students to maintain motivation and to accomplish their achievement. The final conclusion is addressed to the benefits of additional workshops employment in the course, which confirmhigherfinal scores-grades, achievement of creative results (facultative tasks) and confirmation of DG knowledge adaption

The contemporary visualization and modelling technologies and the techniques for the design of the green roofs

The contemporary design solutions are merging the boundaries between real and virtual world. The Landscape architecture like the other interdisciplinary field stepped in a contemporary technologies area focused on that, beside the good execution of works, designer solutions has to be more realistic and “touchable”. The opportunities provided by Virtual Reality are certainly not negligible, it is common knowledge that the designs in the world are already presented in this way so the Virtual Reality increasingly used. Following the example of the application of virtual reality in landscape architecture, this paper deals with proposals for the use of virtual reality in landscape architecture so that designers, clients and users would have a virtual sense of scope e.g. rooftop garden, urban areas, parks, roads, etc. It is a programming language that creates a series of images creating a whole, so certain parts can be controlled or even modified in VR. Virtual reality today requires a specific gadget, such as Occulus, HTC Vive, Samsung Gear VR and similar. The aim of this paper is to acquire new theoretical and practical knowledge in the interdisciplinary field of virtual reality, the ability to display using virtual reality methods, and to present through a brief overview the plant species used in the design and construction of an intensive roof garden in a Mediterranean climate, the basic characteristics of roofing gardens as well as the benefits they carry. Virtual and augmented reality as technology is a very powerful tool for landscape architects, when modeling roof gardens, parks, and urban areas. One of the most popular technologies used by landscape architects is Google Tilt Brush, which enables fast modeling. The Google Tilt Brush VR app allows modeling in three-dimensional virtual space using a palette to work with the use of a three dimensional brush. The terms of two "programmed" realities - virtual reality and augmented reality - are often confused. One thing they have in common, though, is VRML - Virtual Reality Modeling Language. In this paper are shown the ways on which this issue can be solved and by the way, get closer the term of Virtual Reality (VR), also all the opportunities which the Virtual reality offered us. As well, in this paper are shown the conditions of Mediterranean climate, the conceptual solution and the plant species which will be used by execution of intensive green roof on the motel “Marković”