Verify that the Euler's formula is correct?

Question

Theodore Abad · Accepted Answer

Proof by induction below:
(it is long, but not very complicated)

It is necessary to define the environment before we can prove Euler's Formula for Planar Graphs. For the purposes of this discussion, a Planar Graph is: consists of three different kinds of elements: Additionally Common Variables: #V#: the number of vertices in a planar graph. #E#: the number of edges in a planar graph. #F#: the number of faces in a planar graph. Euler's Formula for Planar Graphs: #color(white)("XXX")V-E+F=2# ด ฟฟ ฟ ฟ Considering the aforementioned: One vertex, one edge (with both ends connected to the vertex), and two faces—one inside the loop the edge creates when it loops back to the vertex and the other outside—are the components of a minimal planar graph. Let us suppose that Euler's Formula is true for #color(white)("XXX")V <= n_v#, and #color(white)("XXX")E <= n_e#, and #color(white)("XXX")F <= n_f# Option 1: Adding a Vertex Consider generation of a planar graph with #n_v+1# vertices from some planar graph that satisfies Euler's Formula. We know that prior to adding the #(n_v+1)^("st")# vertex, the prior graph must satisfy Euler's formula. There are two possibilities: - The new vertex is not on an existing edge. In this case a new edge must also be added (see discussion of composition above). That is if we add such a vertex, #V# will increase by #1# but so must #E# and Euler's Formula continues to be valid. - The new vertex is on an existing edge. In this case #V# is increased by #1# but an existing edge is split into two edges, so that #E# also increases by #1# (and Euler's Formula remains valid). Option 2: Adding an Edge Without Adding a Vertex Again there are two possibilities: - Both end of the new edge are connected to the same vertex. In this case we have a loop which encloses a new face within an existing face. Adding a new edge in this way increases #E# by #1# but it also increases #F# by #1# so Euler's Formula remains valid. Note that Option 3—Adding a Face Without Adding a Vertex or Edge—is not available.

Caroline Aucoin · Answer

(see below)

I have taken that you really did mean "verify" which implies that there should have been a sample to verify. It is also possible that you really meant "prove"; if so re-submit this question with proper terminology.

I have also assumed that, since this was asked under the Geometry topic, you meant Euler's Formula relating Vertices, Edges, and Faces for planar figures in Euclidean space (Euler has several formulae in different areas).

Here is a sample for verification:
#7# Vertices;
#9# Edges; and
#4# Faces.

Euler's Formula says
#color(white)("XXX")V-E+F=2#
and for the sample (above) since
#color(white)("XXX")7-9+4=2#
we have verified Euler's Formula (for this example).