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On modular chromatic indexes of graphs. (English) Zbl 1251.05057
Summary: For a connected graph $$G$$ of order 3 or more and an edge coloring $$c: E(G)\to\mathbb{Z}_k$$ ($$k\geq 2$$) where adjacent edges may be colored the same, the color sum $$s(v)$$ of a vertex $$v$$ of $$G$$ is the sum in $$\mathbb{Z}_k$$ of the colors of the edges incident with $$v$$. The edge coloring $$c$$ is a modular $$k$$-edge coloring of $$G$$ if $$s(u)\neq s(v)$$ in $$\mathbb{Z}_k$$ for all pairs $$u$$, $$v$$ of adjacent vertices in $$G$$.
The modular chromatic index $$\chi_m'(G)$$ of $$G$$ is the minimum $$k$$ for which $$G$$ has a modular $$k$$-edge coloring. It is shown that $$\chi(G)\leq\chi_m'(G)\leq\chi(G)+ 1$$ for every connected graph $$G$$ of order at least 3, where $$\chi(G)$$ is the chromatic number of $$G$$. Furthermore, it is shown that $$\chi_m'(G)= \chi(G)+ 1$$ if and only if $$\chi(G) \equiv 2 \pmod 4$$ and every proper $$\chi(G)$$-coloring of $$G$$ results in color classes of odd size.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
connected graph; edge coloring; modular $$k$$-edge coloring