Modular edge colorings of Mycielskian graphs

Abstract

Let $G$ be a connected graph of order $3$ or more and $c:E(G)rightarrowmathbb{Z}_k$‎ ‎($kge 2$) 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 $Vcup V'cup{u},$ where $V'={v':vin V},$ and edge set $Ecup{xy':xyin E}cup{v'u:v'in V'}.$ It is shown that $chi'_m(mathscr{M}(G)),=,chi(mathscr{M}(G))$ for some bipartite graphs‎, ‎cycles and complete graphs‎.