I suppose I should have asked two questions 1)Are all planar Hamiltonian graphs 4 colorable ? 2)If so, then are does this imply all planar graphs are 4 colorable? Then the first paragraph is about reducing Hamiltonian planar graphs to compositions of trees which reduces the first question to the mutual edge colorability of any two trees. (This presupposes you know Taits theorem that 4 coloring faces is equivalent to 3 coloring the edges.) The second question I have not made any real headway on, so I have no discussion , only the question.

thank you for reading my query. Please advise me on what is unclear.? Isnt the question "does the colorability of planar Hamiltonian graphs imply all planar graphs are colorable" clear? Shall I capitalize all beginings of sentences?