A Note on Circular Chromatic Number of Graphs with Large Girth and Similar Problems

Abstract

In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most 5, and for the pth chromatic number χp, from which follows in particular that graphs G of large girth excluding a minor have χp(G)≤p+2.