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 $k\geq 2$ 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 $c(v) = \sum_{e\in E_v} s(e) \mbox{ in } \mathbb{Z}_k,$ (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 $\chi'_{it}(G)$. In this paper, we show that if $G$ is a graph with vertex chromatic number $\chi(G)$, then $\chi'_{it}(G)=\chi(G)$, unless $\chi(G)=2 \pmod 4$ and in this case $\chi'_{it}(G)\in \{\chi(G), \chi(G)+1\}$. Moreover, we show that it is NP-hard to decide whether $\chi'_{it}(G)=\chi(G)$ or $\chi'_{it}(G)=\chi(G)+1$ and give some examples of perfect graph classes for which the problem is polynomial.