Hamiltonian cycles in a graph of degree 4

PROOF OF THEOREM l: The nodes of G may be relabeled so that the Hamiltonian cycle/-/1 is a polygon with branches PoPa, PaPz ..... P.-aPo, and the Hamiltonian cycle Hz consists of chords o f / / 1 . If n is even, a cubic graph Ga containing the Hamiltonian cycle//2 is obtained from G by deleting alternate branches of/ /1 9 Applying Corollary 3 to G~ gives the desired cycle Ha. I fn is odd there are two cases: (1) There is some node/ '1 (say) such that its neighbors Po and P2 on the polygon Ha are not adjacent in/-/2, or (2) for every i the interior Hamiltonian cycle//2 contains a branch Pi-~P~+~, (with subscripts taken modulo n).