Unique Fulkerson coloring of Petersen minor-free cubic graphs

Let G be a cubic graph and the graph 2G is obtained by replacing each edge of G with a pair of parallel edges. A proper 6-edge-coloring of 2G is called a Fulkerson coloring of G. It was conjectured by Fulkerson that every bridgeless cubic graph has a Fulkerson coloring. In this paper we show that for a Petersen-minor free Graph G, G is uniquely Fulkerson colorable if and only if G constructed from K4 via a series of Y−Δ-operations (expending a vertex by a triangle). This theorem is a partial result to the conjecture that, for a Petersen-minor free Graph G, G is uniquely 3-edge-colorable if and only if G constructed from K4 via a series of Y−Δ-operations (expending a vertex by a triangle).