# Extending perfect matchings into Hamiltonian cycles

Let $$G$$ be a simple cubic graph which has a Hamiltonian circuit $$C$$. In general, it is not possible to find a second Hamiltonian circuit which contains all the chords of $$C$$. For example, the Wagner graph with $$C$$ being the outer circuit in the picture contains no such second Hamiltonian circuit.

But it is possible to add an additional chord so that the resulting graph contains a second Hamiltonian circuit including all chords, as in the second picture?

Our first question is:

Question 1 (EGME) Let $$G$$ be a cubic graph which has a Hamiltonian circuit $$C$$ with a set of chords $$D$$. Is it always possible to add a chord $$f$$ of $$C$$ in $$G$$ so that the resulting graph has a Hamiltonian circuit containing all the chords (including the new chord $$f$$)? Or equivalently, is there a Hamiltonian path containing all the chords in D? Please provide proof or counterexample.

Our second question is

Question 2 (EGME) If we add ANY new set of chords of $$C$$ in $$G$$ (as in question 1) which are a perfect matching in the resulting graph (which should be simple), is there a second Hamiltonian circuit which contains all of the chords in the original set? Or equivalently, let $$G$$ be a 4-regular Hamiltonian graph $$G$$ with a Hamiltonian circuit $$C$$, and such that the chords of $$C$$ can be partitioned into two perfect matchings $$P$$, $$Q$$. Is there a Hamiltonian circuit of $$G$$ which contains all the edges in $$P$$ ($$Q$$). If not, under what conditions is this true?

Computer experiments seem to suggest this might be the case.