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Complexity of the improper twin edge coloring of graphs. (English) Zbl 1371.05067
Summary: 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 \(c(v) = \sum_{e\in E_v} s(e)\) 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 \(\chi^\prime_{it}(G)\). It is known that \(\chi^\prime_{it}(G)=\chi (G)\), unless \(\chi (G) \equiv 2 \pmod 4\) and in this case \(\chi^\prime_{it}(G)\in \{\chi (G), \chi (G)+1\}\). 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 \(t\geq k\); 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 \(\chi^\prime_{it}(G)=\chi (G)\) or \(\chi^\prime_{it}(G)=\chi (G)+1\), 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.
Reviewer: Reviewer (Berlin)
MSC:
05C15 Coloring of graphs and hypergraphs
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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