Equitable Coloring of Mycielskian of Some Graphs

In a search for triangle-free graphs with arbitrarily large chromatic numbers, mycielski developed a graph transformation that transforms a graph g into a new graph µ(g), we now call the mycielskian of g, which has the same clique number as g and whose chromatic number equals χ(g) + 1. this paper presents exact values of the equitable chromatic number χ= for the mycielski’s graph of complete graphs µ(kn), the mycielski’s graph of cycles µ(cn), the mycielski’s graph of paths µ(pn), the mycielski’s graph of helm graphs µ(hn) and the mycielski’s graph of gear graphs µ(gn).