Weak oddness as an approximation of oddness and resistance in cubic graphs

We consider the question whether the least number ω of odd components in a 2-factor is the same as the least number ωw of odd components in an even factor of a cubic graph. Both invariants appear in the literature under the same name “oddness”. We show that these two invariants are different by constructing graphs G satisfying ρ(G)<ωw(G)<ω(G) (where ρ denotes resistance, i.e. the least number of edges that have to be removed from a cubic graph in order to turn it into a 3-edge-colourable graph). In addition, we demonstrate that the difference between any two of those three invariants may be arbitrarily large. Our results imply that if we replace a vertex of a cubic graph with a triangle, then its oddness (ω) may significantly decrease.