#Chromatic number , Planarity

Question

Let G be a planar graph such that every face is bordered by exactly 3 edges.Which of the following can never be the value for χ(G) ? (where χ(G) is the chromatic number of G)

a)  2

b)  3

c)  4

d)  None of these

PS : (Explain: "every face is bordered by exactly 3 edges. ")

Comments

"every face is bordered by exactly 3 edges. "

There must be a cycle of length 3, so it cannot be 2 colourable.

Answer 1

I think answer should be (A) because K₂ require 2 colors and is not bounded by three edges but K₃ require 3 colors, and K₄ require 4 colors to color them and both of them are planar graphs and are bounded by exactly 3 edges. Moreover, every face is bordered by exactly three edges means that every REGION formed within the graph must have exactly three edges surrounding them. Have a look at the graphs, every region within graph K₃ and K₄ are bounded by exactly three edges but K₂ is not.

Answer 2

as it mentions it's planar and every face is bounded by exactly three edges so the only possibility we have K3 which has chromatic number X(G) =3