The structure of neighborhoods of 5-vertices in normal plane maps with minimum degree 5

In 1940, Lebesgue described the neighborhoods of vertices of degree 5 in normal plane maps with minimum degree 5 (M5), presenting only an idea of the proof but not the details. The paper presents a detailed scheme of a complete proof of Lebesgue's description with improving two of its parameters without worsening the others. Moreover, it is present a scheme of the proof of the height of a 5-star (the maximum degree of its vertices) in an M5, which improves the result of O.V.Borodin, A.O.Ivanova, T.R.Yensen (2013).