Euler's formula and promotion of China Unicom's floor plan
Euler's formula for China Unicom's floor plan
The number of regions divided by the plan of China Unicom + the number of vertices-the number of boundaries = 2
R is the number of regions, V is the number of vertices (the order of the graph), and E is the number of boundaries, then R+ V- E= 2
Proof by induction: When E = 0, R = 1, V = 1, R + V − E = 2. Suppose when E = k, R + V − E = 2 holds E = k + 1 {No loop (tree): add an edge and no loop is formed, so a leaf R + (V + 1) − (E + 1) = 2 There is a loop: an edge is added and a loop is formed, so an area is also added (R + 1) + V − (E + 1) = 2 \Large Proof of induction: \\ \normalsize when E=0 When, R=1,V=1,R+V-E=2. Suppose when E=k, R+ V- E= 2 holds \\ E=k+1\left\{\begin{array}{l}\mathrm{no loop}(tree): \mathrm{add an edge and No loop is formed, so a leaf is added)R+(V+1)-(E+1)=2\\\mathrm(with loop)\;\;\;\;\;\;:\;\;\ mathrm(adding an edge and forming a loop, so also adding an area)\;(R+1)+V-(E+1)=2\end(array)\right.Owned satisfied law of evidence out :This E=0 o'clock , R=1,V=1,R+V−E=2 . Set when E=k When , R & lt+V−E=2 to standE=k+1{ No return path ( tree ) : by adding a Article sides and and did not have a form to back path is to be increased plus the a sheet leaf R+( V+1)−(E+1)=2There are back road:By adding a Article sides and and formed into a return path is to be increased add the a th region domain(R+1)+V−(E+1)=2
The promotion of Euler's formula of China Unicom's floor plan:
Proof of K5 and K3,3 non-planar drawings
K5 regular graph, the regular graph with the degree of each vertex being 5
V5 vertices, E10 edges, E≤3V-6, 10 is not less than or equal to 9
K3,3, the two parts are bipartite graphs with three vertices
The minimum number of times to form a surface is 4. At least four edges of four vertices are needed to divide the plane into a region, so the number of edges is L=4, E≤L * (V-2)/(L-2), 9 is not Less than or equal to 8