3-consecutive edge coloring of a graph

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