(2 + ?)-Coloring of planar graphs with large odd-girth

Abstract

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function ƒ(ϵ) for each ϵ : 0 < ϵ < 1 such that, if the odd-girth of a planar graph G is at least ƒ(ϵ), then G is (2 + ϵ)-colorable. Note that the function ƒ(ϵ) is independent of the graph G and ϵ → 0 if and only if ƒ(ϵ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109–119, 2000