Avoiding 7-circuits in 2-factors of cubic graphs

Let G be a cyclically 4-edge-connected cubic graph with girth at least 7 on n vertices. We show that G has a 2-factor F such that at least a linear amount of vertices is not in 7-circuits of F. More precisely, there are at least n/657 vertices of G that are not in 7-circuits of F. If G is cyclically 6-edge-connected then the bound can be improved to n/189. As a corollary we obtain bounds on the oddness and on the length of the shortest travelling salesman tour in a cyclically 4-edge-connected (6-edge-connected) cubic graph of girth at least 7.