Textbook: Introduction to Graph Theory by Robert J. Wilson 5th Edition Pg. 91

4.185 (i) Let Gbe a simple connected cubic plane graph, and let C, be the number of k-sided faces. By counting the number of vertices and edges of G, prove that 3C + 2C + Cs - G-2C -3Cq - . . .= 12. (ii) Use this result to deduce the result of Exercise 4.16(ii). (iii) Deduce also that G has at least one face bounded by at most five edges.4.18 (i) If G has nvertices, m edges and ffaces, then f=CG+C+C+CG+. .., 2m=3C,+4C,+5C,+6C+.. ., 3n=3G+4C+5CG+6G +. ... Substituting these expressions for m and n into Euler's formula yields the result. (ii) Since G=C,=G=G=...=0, we deduce that C; = 12. (iii) If G 'has no face bounded by at most five edges, then (; = C, = C; =0, and the lefi- hand side is negative; this is a contradiction