Improper Twin Edge Coloring of Graphs

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex coloring of G, then we refer to such a coloring as an improper twin k-edge coloring. The minimum k for which G has an improper twin k-edge coloring is called the improper twin chromatic index of G and is denoted by χit(G). In this paper, we show that if G is a graph with vertex chromatic number χ(G), then χit(G) = χ(G), unless χ(G) = 2 (mod 4) and in this case χit(G) ∈ {χ(G), χ(G) + 1}. Moreover, we show that it is NP-hard to decide whether χit(G) = χ(G) or χ ′ it(G) = χ(G) + 1 and give some examples of perfect graph classes for which the problem is polynomial.