modular edge colorings of mycielskian graphs

Let g be a connected graph of order 3 or more and c : e(g) → zk (k ≥ 2) a k-edge coloring of g where adjacent edges may be colored the same. the color sum s(v) of a vertex v of g is the sum in zk of the colors of the edges incident with v. the k-edge coloring c is a modular k-edge coloring of g if s(u) ̸= s(v) in zk for all pairs u, v of adjacent vertices of g. the modular chromatic index χ′ (g) of g is the minimum k for which g has a modular k-edge coloring. the mycielskian of g = (v, e) is the graph m (g) with vertex set v ∪ v ′ ∪ {u}, where v ′ = {v′ : v ∈ v }, and edge set e ∪ {xy′ : xy ∈ e} ∪ {v′u : v′ ∈ v ′}. it is shown that χ′ m (m (g)) = χ(m (g)) for some bipartite graphs, cycles and complete graphs.