we have a simple graph with n vertices and hamiltonian cycle, prove it has 2 even subgraphs T,S such as each edge of G is at least in one of T or S.
part 2:
if n is even , prove it has 2 odd subgraphs T2 , S2 such as each edge of G is at least in one of T2 or S2.
my attempt:
i think this condition holds for this as well: every bridgeless graph with n vertices is the union of 2 even graphs
and every edge in graph with hamiltonian cycle is in a cycle
if we prove this the problem is solved but I'm stuck