A METHOD TO OBTAIN LOWER BOUNDS FOR CIRCULAR CHROMATIC NUMBER

Abstract

The circular chromatic number $\chi_c(G)$ of a graph $G$ is a very natural generalization of the concept of chromatic number $\chi(G)$, and has been studied extensively in the past decade. In this paper we present a new method for bounding the circular chromatic number from below. Let $\omega$ be an acyclic orientation of a graph $G$. A sequence of acyclic orientations $\omega_1$, $\omega_2$, $\omega_3,\ldots$ is obtained from $\omega$ in such a way that $\omega_1=\omega$, and $\omega_i$ ($i\geq 2$) is obtained from $\omega_{i-1}$ by reversing the orientations of the edges incident to the sinks of $w_{i-1}$. This sequence is completely determined by $\omega$, and it can be proved that there are positive integers $p$ and $M$ such that $\omega_i=\omega_{i+p}$ for every integer $i\geq M$. The value $p$ at its minimum is denoted by $p_\omega$. To bound $\chi_c(G)$ from below, the methodology we develop in this paper is based on the acyclic orientations $\omega_M, \omega_{M+1},\cdots,\omega_{M+p_\omega-1}$ of $G$. Our method demonstrates for the first time the possibility of extracting some information about $\chi_c(G)$ from the period $\omega_M, \omega_{M+1},\cdots,\omega_{M+p_\omega-1}$ to derive lower bounds for $\chi_c(G)$.