Complexity of the Improper Twin Edge Coloring of Graphs

Abstract

Let G be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow {\mathbb {Z}}_k$ for some integer $k\ge 2$ be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring $c : V (G) \rightarrow {\mathbb {Z}}_k$ defined by $c(v) = \sum _{e\in E_v} s(e) \text{ in } {\mathbb {Z}}_k,$ (where the indicated sum is computed in ${\mathbb {Z}}_k$ and $E_v$ 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 $\chi '_{it}(G)$. It is known that $\chi '_{it}(G)=\chi (G)$, unless $\chi (G) \equiv 2 \pmod 4$ and in this case $\chi '_{it}(G)\in \{\chi (G), \chi (G)+1\}$. 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 $t\ge k$; we call this the monotonicity property. In addition, we provide a linear time algorithm to construct an improper twin edge coloring using at most $k+1$ 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 $\chi '_{it}(G)=\chi (G)$ or $\chi '_{it}(G)=\chi (G)+1$, 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.