On the edge-coloring problem for a class of 4-regular maps

A (plane) 4-regular map G is called C-simple if it arises as a superposition of simple closed curves (tangencies are not allowed); in this case σ (G) is the smallest integer k such that the curves of G can be colored with k colors in such a way that no two curves of the same color intersect. We prove that if σ (G) ≤ 4, G is edge colorable with 4 colors. Moreover we show that a similar result for maps G with σ(G) ≤ 5 would imply the Four-Color Theorem.