A Fullerene is a 3regular planar graph with faces of degree

A Fullerene is a 3-regular planar graph with faces of degree 5 and 6. Let P be the number of pentagons (degree 5 faces) and H be the number of hexagons (degree 6 faces). Given that a Fullerene has v vertices, determine P and H as functions of v.

Solution

Here there are P faces of degree five( pentagons ) and H faces of degree six( Hexagons)

Then there are F = P+H faces

we know from eulers formula that F= 2 + E - V

By 3 Regularity we know 2E = 3V

so P+H = F = 2 + 3V/2 -V

= 2 + V/2

Further by handshaking for planar graphs we know

2E = 5P + 6H

so , 3V= 2 E = 5P + 6H

= 5(P+H) + H

= 5(2+ v/2) + H

This tells us that

v = 2H + 20,

so H = V/2 - 10

as P + H = 2+V/2

we conclude

P = 2+ V/2 - H

= 2+ V/2 - (V/2 - 10)

= 12