The Circular Chromatic Number of the Mycielskian of Mt(Kn)

TL;DR

This paper investigates the circular chromatic number of iterated Mycielski graphs of complete graphs, proving a strengthened condition under which it equals the chromatic number, extending previous conjectures and results.

Contribution

The paper extends existing results by establishing a broader condition for the equality of circular and chromatic numbers in Mycielski graphs of complete graphs.

Findings

Proves that for t ≥ 4 and sufficiently large n, the circular chromatic number equals the chromatic number.

Strengthens previous conjectures by reducing the bounds on n for the equality to hold.

Provides new insights into the chromatic properties of iterated Mycielski graphs.

Abstract

As a natural generalization of chromatic number of a graph, the circular chromatic number of graphs (or the star chromatic number) was introduced by A.Vince in 1988. Let $M^t(G)$ denote the $t$th iterated Mycielski graph of $G$. It was conjectured by Chang, Huang and Zhu(Discrete mathematics,205(1999), 23-37) that for all $n \ge t+2, \chi_c(M^t(K_n))=\chi(M^t(K_n))=n+t.$ In 2004, D.D.F. Liu proved the conjecture when $t\ge 2$, $n\ge 2^{t-1}+2t-2$. In this paper,we show that the result can be strengthened to the following: if $t\ge 4$, $n\ge {11/12}2^{t-1}+2t+{1/3}$, then $\chi_c(M^t(K_n))=\chi(M^t(K_n))$.