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An upper bound for the twin chromatic index of a graph. (English) Zbl 1314.05038
Summary: For a connected graph $$G$$ of order at least 3 and an integer $$k\geq 2$$, a twin edge $$k$$-coloring of $$G$$ is a proper edge coloring of $$G$$ with the elements of $$\mathbb{Z}_k$$ so that the induced vertex coloring in which the color of a vertex $$v$$ in $$G$$ is the sum (in $$\mathbb{Z}_k$$) of the colors of the edges incident with $$v$$ is a proper vertex coloring. The minimum $$k$$ for which $$G$$ has a twin edge $$k$$-coloring is called the twin chromatic index of $$G$$ and is denoted by $$\chi_t'(G)$$.
In this note, we show that $$\chi_t'(G)\leq< 4\Delta(G)-3$$ for every connected graph $$G$$ of order at least 3.

##### MSC:
 05C05 Trees 05C15 Coloring of graphs and hypergraphs 05C40 Connectivity
##### Keywords:
edge and vertex coloring; twin edge coloring