Consider a graph of order n where n is 1 mod 4 (I.E. pentagons, nonagons, etc.), and suppose it is a (n-1)/2-regular graph. Also (potentially optionally) suppose that both it and its complement are connected.
Can this type of graph be verified to be Hamiltonian, and if so, how would the proof roughly go?
As per Ore's theorem
A graph with n vertices (n ≥ 3) is Hamiltonian if, for every pair of non-adjacent vertices, the sum of their degrees is n or greater
In your case, it is (n-1)/2 regular graph. Therefore sum of non-adjacent vertices will be (n-1).
Also, you can use the following theorem by Tutte(1956):
A 4-connected planar graph has a Hamiltonian cycle.
As per my opinion, if there a 4-connected planar graph is sub-graph on all the vertices of the given graph, then Hamiltonian cycle will exists.
