The Four-Color Theorem originated from the attempt to solve the problem of painting MAPS over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until it was resolved in 1976. The proposed solution, which is disruptive, computationally calculates the number of possible states for a representation of a flat map. Although it is resolved, the lack of a formal proof for this problem causes some discomfort. Therefore, a solution that uses more traditional techniques and is easily understandable is needed. In a previous article, a solution based on equalities and inequalities between boundaries was presented. Now in this article, a generic spheroidal MAP is subjected to various one-to-one relationships until a generator of all possible MAPS on a two-dimensional surface partitioned into n2 cells are found. Four-Colors are proved to be necessary and sufficient to paint a two dimensional MAP. It is explained at the end of the article that the imposition of a fifth color as a necessary condition implies a contradiction.

Com a globalização, crescem os níveis de rivalidade e pressão competitiva entre os concorrentes atuando nos mesmos mercados e áreas geográficas. Mesmo assim, as pequenas e médias empresas vêm se tornando participantes críticos nas economias nacionais e nos negócios mundiais. Mas pouco se sabe sobre quais estratégias de operações estas pequenas e médias empresas utilizam diante da globalização. No Brasil, com o aumento dos processos de fusões, aquisições que se articulam entre as organizações globais e empresas que atuam nacionalmente, aumentaram a turbulência e a incerteza nos mercados. As empresas que conseguem obter um bom conhecimento do ambiente em que se inserem podem tomar ações pró-ativas e influenciar na evolução do sistema. Este artigo trata da relação entre ambiente de competição, incertezas, ameaças e oportunidades percebidas, com foco em pequenas e médias empresas. É apresentado um estudo de caso de uma empresa de médio porte, denominada de X, que possuía uma situação estável no mercado por aproximadamente onze anos, ocupando a segunda posição, até que se deu a aquisição da empresa líder nacional pela líder global. O artigo apresenta as sugestões seguidas e que auxiliaram esta empresa a manter-se competitiva, embora carente de recursos financeiros e humanos.

Globalization has led to increasing levels of rivalry and competitive pressure among competitors acting in the same markets and geographical regions. Nonetheless, small and medium enterprises are becoming critical players in national economies and worldwide businesses. However, little is known about the operations strategies these SMEs employ to face the pressures of globalization. In Brazil, the growing market upheaval and uncertainty has been aggravated by the increasing number of mergers with and acquisitions by global organizations and companies operating in the country. In this context, companies that are knowledgeable about the environment in which they operate can take proactive action and influence the system's evolution. This article discusses the relations between competitive business environments, perceived uncertainties, threats and opportunities from the standpoint of SMEs. A case study of a medium size company, dubbed X, is described here. This company enjoyed a stable situation in the market for about eleven years, ranking second in its area in the domestic market, until the domestic leader company was acquired by a leading global corporation. This article lists and discusses suggestions that helped the company remain competitive despite its paucity of financial and human resources.