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

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 f( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f( ), then G is (2 + )-colorable. Note that the function f( ) is independent of the graph G and → 0 if and only if f( )→∞. 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. c © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109–119, 2000