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(2 + ϵ)‐Coloring of planar graphs with large odd‐girth
Author(s) -
Klostermeyer William,
Zhang Cun Quan
Publication year - 2000
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(200002)33:2<109::aid-jgt5>3.0.co;2-f
Subject(s) - combinatorics , mathematics , lemma (botany) , girth (graph theory) , discrete mathematics , planar graph , graph coloring , triangle free graph , graph , odd graph , graph power , chordal graph , 1 planar graph , line graph , ecology , poaceae , biology
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