Multi-coloring and Mycielski’s construction

We consider a number of related results taken from two papers – one by W. Lin [1], and the other D. C. Fisher[2]. These articles treat various forms of graph colorings and the famous Mycielski construction. Recall that the Mycielskian μ(G) of a simple graph G is a graph whose chromatic number satisfies χ(μ(G)) = χ(G)+1, but whose largest clique is no larger than the largest clique in G. Extending the work of Mycielski, the results presented here investigate how the Mycielski construction affects a related parameter called the k chromatic number χk(G), developing an upper and lower bound for this parameter when applied to μ(G). We then prove that there are infinite families of graphs that realize both the upper and lower upper bounds. Alongside these main results, we also include a remarkable curiosity, first found by Fisher, that although the fractional chromatic number of a graph G might be expressible as a/b in lowest terms, that does not necessarily imply that there exists a proper coloring of G with the belement subsets of an a-element set. We also demonstrate that the ratio χk(G)/k, where χk(G) denotes the k-tuple chromatic number, is not a strictly decreasing function.