# Updates to proofs of Euler's formula

I added two new proofs to my page of proofs of Euler's formula \( V-E+F=2 \) for planar graphs, following suggestions from Matthias Beck. One proof works by defining a valuation on unions of cells of a hyperplane arrangement, and embedding a given convex polytope within a hyperplane arrangement in one higher dimension in such a way that inclusion-exclusion may be easily used; the other proof involves polynomials that count the number of integer points in a scaled copy of a shape. These integer-point-counting polynomials are studied in much greater detail in Beck's book with Robins on integer point enumeration, a preliminary version of which is available free online.