(a) Show that a planar graph G has a 2-face-colouring if and only if G is Eulerian. (Show that the dual of G is bipartite and that any bipartite graph has an Eulerian dual.) (b) Show that every planar...

1. 
   

   
   (a) Show that a planar graph
   
   G
   
   has a 2-face-colouring if and only if
   

   
   

G

is Eulerian.

(Show that the

dual



of

G



is bipartite and that any bipartite graph has an Eulerian dual.)

(b) Show that every planar Hamiltonian graph has a 4-face-colouring. (Any Hamiltonian circuit divides the plane into two regions. Consider using two colours for the faces in either region.)