Berge–Fulkerson coloring for C₍₁₂₎-linked permutation graphs

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Let (Formula presented.) be a permutation graph with a 2-factor (Formula presented.). A circuit (Formula presented.) is (Formula presented.) -alternating if (Formula presented.) is a perfect matching of (Formula presented.). A permutation graph (Formula presented.) with a 2-factor (Formula presented.) is (Formula presented.) -linked if it contains an (Formula presented.) -alternating circuit of length at most 12. It is proved in this paper that every (Formula presented.) -linked permutation graph is Berge–Fulkerson colorable. As an application, the conjecture is verified for some families of snarks constructed by Abreu et al., Brinkmann et al., and Hägglund et al.