Modular edge colorings of Mycielskian graphs

Let G be a connected graph of order ۳ or more and c:E(G)\rightarrow\mathbb{Z}_k‎ ‎(k\ge ۲) a k-edge coloring of G 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 k-edge coloring c is a modular k-edge coloring of G if s(u)\ne s(v) in \mathbb{Z}_k for all pairs u, v of adjacent vertices of G. The modular chromatic index \chi'_m(G) of G is the minimum k for which G has a modular k-edge coloring‎. ‎The Mycielskian of G\,=\,(V,E) is the graph \mathscr{M}(G) with vertex set V\cup V'\cup\{u\}, where V'=\{v':v\in V\}, and edge set E\cup\{xy':xy\in E\}\cup\{v'u:v'\in V'\}. It is shown that \chi'_m(\mathscr{M}(G))\,=\,\chi(\mathscr{M}(G)) for some bipartite graphs‎, ‎cycles and complete graphs‎.

Keywords:

modular edge coloring , modular chromatic index , Mycielskian of a graph