Problems with Cryptography, Polynomials, Graphs, Recursive Functions, and Message Errors

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1. First of all we prove that in any connected planar graph it holds the following inequality: e≤3*v-6.
Sum of the degrees for faces is twice the number of edges; also each face has the degree at least 3. From these findings => 2*e≥3*f .
But Euler’s formula => v-e+f=2 => f = e-v+2 => 3f=3e-3v+6 and using the above result=> 2e≥3e-3v+6 => 3v≥ e+6....

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