Interval incidence coloring of subcubic graphs

For a given simple graph G = (V,E), we define an incidence as a pair (v, e), where vertex v ∈ V (G) is one of the ends of edge e ∈ E(G). Let us define a set of incidences I(G) = {(v, e) : v ∈ V (G)∧ e ∈ E(G)∧ v ∈ e}. We say that two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, e 6= f , (ii) e = f , v 6= w, (iii) e = {v, w}, f = {w, u} and v 6= u. By an incidence coloring of G we mean a function c : I(G)→ N such that c((v, e)) 6= c((w, f)) for any adjacent incidences (v, e) and (w, f) [1]. By an interval incidence coloring we mean an incidence coloring such that the set of colors of incidences adjoining the same vertex forms an interval, and by interval incidence coloring number χii we mean the smallest number of colors in any interval incidence coloring. This problem was studied in [2, 3]. The open question is whether χii(G) ≤ 2∆ in general case [3]. We solved this problem for any subcubic graph G, i.e. we proved χii(G) ≤ 6.