3-consecutive edge coloring of a graph

Three edges e1e1, e2e2 and e3e3 in a graph GG are consecutive if they form a path (in this order) or a cycle of length 3. The 3-consecutive edge coloring number  ψ3c′(G) of GG is the maximum number of colors permitted in a coloring of the edges of GG such that if e1e1, e2e2 and e3e3 are consecutive edges in GG, then e1e1 or e3e3 receives the color of e2e2. Here we initiate the study of ψ3c′(G).A close relation between 3  -consecutive edge colorings and a certain kind of vertex cut is pointed out, and general bounds on ψ3c′ are given in terms of other graph invariants. Algorithmically, the distinction between ψ3c′=1 and ψ3c′=2 is proved to be intractable, while efficient algorithms are designed for some particular graph classes.