Discrete structures 2-region graph

Consider a map with 2n regions which are labeled with integers from 1 to n so that there exactly 2 regions with each number (an example is given below). A 2-region graph of such a map is a graph with vertices {1,,n} where the edge {v,w} is part of the graph if and only if there is a region with label v that shares some boundary piece with a region with label w in the map. In this exercise we will prove that any 2-region graph has chromatic number at most 12. Note: In this exercise you can use the fact that the dual graph of a map as explained in the lecture is a planar graph. Consider how the normal dual graph is related to the 2-region graph of a map. 6a Prove that the average degree of a 2-region graph is less than 12