Let G be a graph whose each component has order at least 3. Let s:E(G)→Zk\documentclass[12pt]{minimal}
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\begin{document}$$s : E(G) \rightarrow {\mathbb {Z}}_k$$\end{document} for some integer k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$c : V (G) \rightarrow {\mathbb {Z}}_k$$\end{document} defined by c(v)=∑e∈Evs(e)inZk,\documentclass[12pt]{minimal}
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\begin{document}$$c(v) = \sum _{e\in E_v} s(e) \text{ in } {\mathbb {Z}}_k,$$\end{document} (where the indicated sum is computed in Zk\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_k$$\end{document} and Ev\documentclass[12pt]{minimal}
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\begin{document}$$E_v$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$$\chi '_{it}(G)$$\end{document}. It is known that χit′(G)=χ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\chi '_{it}(G)=\chi (G)$$\end{document}, unless χ(G)≡2(mod4)\documentclass[12pt]{minimal}
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\begin{document}$$\chi (G) \equiv 2 \pmod 4$$\end{document} and in this case χit′(G)∈{χ(G),χ(G)+1}\documentclass[12pt]{minimal}
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\begin{document}$$\chi '_{it}(G)\in \{\chi (G), \chi (G)+1\}$$\end{document}. In this paper, we first give a short proof of this result and we show that if G admits an improper twin k-edge coloring for some positive integer k, then G admits an improper twin t-edge coloring for all t≥k\documentclass[12pt]{minimal}
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\begin{document}$$t\ge k$$\end{document}; we call this the monotonicity property. In addition, we provide a linear time algorithm to construct an improper twin edge coloring using at most k+1\documentclass[12pt]{minimal}
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\begin{document}$$k+1$$\end{document} colors, whenever a k-vertex coloring is given. Then we investigate, to the best of our knowledge the first time in literature, the complexity of deciding whether χit′(G)=χ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\chi '_{it}(G)=\chi (G)$$\end{document} or χit′(G)=χ(G)+1\documentclass[12pt]{minimal}
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\begin{document}$$\chi '_{it}(G)=\chi (G)+1$$\end{document}, and we show that, just like in case of the edge chromatic index, it is NP-hard even in some restricted cases. Lastly, we exhibit several classes of graphs for which the problem is polynomial.
