R+ V- E= 2 is Euler's formula.
Extended data
Prove by mathematical induction
(1) When R= 2, from the description of 1, these two regions can be imagined as two hemispheres with the equator as the boundary, and there are two "vertices" on the equator that divide the equator into two "boundaries", that is, R= 2, V= 2 and E = 2；; So R+ V- E= 2, euler theorem is established.
(2) Let R= m(m≥ 2) and euler theorem is established. It is proved that euler theorem also holds when R= m+ 1.
It can be seen from Note 2 that if we select an area X on the map with R= m+ 1, then X must have an area Y adjacent to it, so after removing the unique boundary between X and Y, there are only m areas on the map; After removing the boundaries of X and Y, if the vertices at both ends of the original boundary are still vertices of three or more boundaries, the vertices will remain, while the number of other boundaries will remain unchanged. If the vertex at one or both ends of the original boundary is now the vertex of two boundaries, the vertex is deleted, and the two boundaries on both sides of the vertex become one boundary. Therefore, when removing the unique boundary between x and y, there are only three situations:
(1) Reduce the area and boundary;
② Reduce an area, a vertex and two boundaries;
③ Reduce one area, two vertices and three boundaries;
That is, when removing the boundary between X and Y, there must be "number of reduced regions+number of reduced vertices = number of reduced boundaries" in any case. We reverse the above process (that is, draw the boundary between x and y as it is), and it will become a graph of R= m+ 1. In this process, it must be "increased number of regions+increased number of vertices = increased number of boundaries".
Therefore, if euler theorem holds when R= m (m≥2), then euler theorem holds when R= m+ 1.
According to (1) and (2), euler theorem holds for any positive integer R≥2.
Refer to Euler Formula _ Baidu Encyclopedia?