Suppose G is a graph with n vertices and m edges. Let n¡¬ be the maximum number of vertices in an induced bipartite subgraph of G and let m¡¬ be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G) = m¡¬/m is called the bipartite density of G, and b∗(G) = n¡¬/n is called the bipartite ratio of G. It is proved in [18] that if G is a 2-connected triangle-free subcubic graph, then apart from seven exceptional graphs, we have b(G) ≥ 17/21. If G is a 2-connected triangle-free subcubic graph, then b∗(G) ≥ 5/7 provided that G is not the Petersen graph and not the dodecahedron. These two results are consequences of a more technical result which is proved by induction: If G is a 2-connected triangle-free subcubic graph with minimum degree 2, then G has an induced bipartite subgraph H with |V (H)| ≥ (5n3 + 6n2 + ǫ(G))/7, where ni = ni(G) are the number of degree i vertices of G, and ǫ(G) ∈ {−2,−1, 0, 1}. To determine ǫ(G), four classes of graphs G1, G2, G3 and F-cycles are onstructed. For G ∈ Gi, we have ǫ(G) = −3 + i and for an F-cycle G, we have ǫ(G) = 0. Otherwise, ǫ(G) = 1. To construct these graph classes, eleven graph operations are used. This thesis studies the structural property of graphs in G1, G2, G3. First of all, a computer algorithm is used to generate all the graphs in Gi for i = 1, 2, 3. Let P be the set of 2-edge connected subcubic triangle-free planar graphs with minimum degree 2. Let G¡¬ 1 be the set of graphs in P with all faces of degree 5, G¡¬2 the set of graphs in P with all faces of degree 5 except that one face has degree 7, and G¡¬3 the set of graphs in P with all faces of degree 5 except that either two faces are of degree 7 or one face is of degree 9. By checking the graphs generated by the computer algorithm, it is easy to see that Gi ⊆ G¡¬i for i = 1, 2, 3. The main results of this thesis are that for i = 1, 2, Gi = G¡¬i and G¡¬3 = G3 ¡åR, where R is a set of nine F-cycles.

[pt] O principal objetivo desta dissertação é realizar a previsão da reserva IBNR. Para isto foi desenvolvido um modelo estatístico de distribuições combinadas que busca uma adequada representação dos dados. A reserva IBNR, sigla em inglês para Incurred But Not Reported, representa o montante que as seguradoras precisam ter para pagamentos de sinistros atrasados, que já ocorreram no passado, mas ainda não foram avisados à seguradora até a data presente. Dada a importância desta reserva, diversos métodos para estimação da reserva IBNR já foram propostos. Um dos métodos mais utilizado pelas seguradoras é o Método Chain Ladder, que se baseia em triângulos run-off, que é o agrupamento dos dados conforme data de ocorrência e aviso de sinistro. No entanto o agrupamento dos dados faz com que informações importantes sejam perdidas. Esta dissertação baseada em outros artigos e trabalhos que consideram o não agrupamento dos dados, propõe uma nova modelagem para os dados não agrupados. O modelo proposto combina a distribuição do atraso no aviso da ocorrência, representada aqui pela distribuição log-normal truncada (pois só há informação até a última data observada); a distribuição da quantidade total de sinistros ocorridos num dado período, modelada pela distribuição Poisson; e a distribuição do número de sinistros ocorridos em um dado período e avisados até a última data observada, que será caracterizada por uma distribuição Binomial. Por fim, a quantidade de sinistros IBNR foi estimada por método e pelo Chain Ladder e avaliou-se a capacidade de previsão de ambos. Apesar da distribuição de atrasos do modelo proposto se adequar bem aos dados, o modelo proposto obteve resultados inferiores ao Chain Ladder em termos de previsão. / [en] The main objective of this dissertation is to predict the IBNR reserve. For this, it was developed a statistical model of combined distributions looking for a new distribution that fits the data well. The IBNR reserve, short for Incurred But Not Reported, represents the amount that insurers need to have to pay for the claims that occurred in the past but have not been reported until the present date. Given the importance of this reserve, several methods for estimating this reserve have been proposed. One of the most used methods for the insurers is the Chain Ladder, which is based on run-off triangles; this is a format of grouping the data according to the occurrence and the reported date. However this format causes the lost of important information. This dissertation, based on other articles and works that consider the data not grouped, proposes a new model for the non-aggregated data. The proposed model combines the delay in the claim report distribution represented by a log normal truncated (because there is only information until the last observed date); the total amount of claims incurred in a given period modeled by a Poisson distribution and the number of claims occurred in a certain period and reported until the last observed date characterized by a binomial distribution. Finally, the IBNR reserve was estimated by this method and by the chain ladder and the prediction capacity of both methods will be evaluated. Although the delay distribution seems to fit the data well, the proposed model obtained inferior results to the Chain Ladder in terms of forecast.