V-E+F = 2
A plane graph is a construction in the plane consisting of finitely many vertices (points) and
finitely many smoothly-drawn, non-crossing edges (lines or curves) connecting some pairs of
them. The vertices and edges divide the plane into connected regions called faces (including one
“outside” face).
Theorem: If G is a connected plane graph with v vertices, e edges, and f faces, and if v ≥ 1,
then
v – e + f = 2.
Example: 5 vertices, 6 edges, 3 faces (counting the infinite one): 5 – 6 + 3 = 2.
Try some more complicated examples.
Problem A: Is the result still true if the graph is not connected? (G is connected if you can get
from any vertex to any other vertex along a sequence of edges.)
Theorem: If G is a connected graph drawn on a sphere, then still v – e + f = 2.
Example: A convex polyhedron can be thought of as a graph on a sphere. A cube
has 8 vertices, 12 edges, and 6 faces; and 8 – 12 + 6 = 2.
Problem B: Suppose that every vertex of G has degree 3. (The degree of a vertex is the number
of edges attached to it.) Can G have exactly 999 vertices?
Problem C: Suppose that every vertex of G has degree 3. Write formulas for e and f in terms of
v only.
Problem D: A map is a connected plane graph in which every vertex has degree at least 3.
Show that the average valence of the faces in any map is less than 6. (The valence of a
face is the number of edges on its boundary.)
Problem E. Explain why soccer balls have 12 pentagonal pieces. (The other pieces are all
hexagons.)
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Problem F. Show that the graph K5 cannot be drawn in a plane without crossings.
(K5 is the graph with five vertices, and a line between every pair of vertices.)
Problem G. Show that the graph K3,3 cannot be drawn in a plane without crossings.
(K3,3 is the graph with vertices u, v, w, x, y, z, with each of u, v, w connected
to each of x, y, z—nine lines in all.)
Problem E (Belarus Mathematical Olympiad, 2000). In a convex polyhedron with m
triangular faces (and possibly some other types of faces too) exactly 4 edges meet at each
vertex. Find the smallest possible value of m.
Problem F (2002 Putnam, problem B2). Consider a polyhedron with at least five faces such
that exactly three edges emerge from each of its vertices. Two players play the following
game:
Each player, in turn, signs his or her name on a previously
unsigned face. The winner is the player who first succeeds in
signing three faces that share a common vertex.
Show that the player who signs first will always win by playing as well as possible.
Problem G. Prove the four color theorem: Using just 4 colors, show that the faces of any map
can be colored in such a way that faces that share an edge have different colors.
If you don’t have time to prove this for four colors, prove it for six colors.
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