Abstract
In Chap. 12 we proved that graphs with girth at least 5, i.e. graphs with no triangles or 4-cycles, have chromatic number at most EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGpbGaaiikamaalaaapaqaa8qacqqHuoara8aabaWdbiaadMea % caWGUbGaeuiLdqeaaiaacMcacaGGUaaaaa!3DCE!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$O(\frac{\Delta }{{In\Delta }}).$$ In this chapter, we will present Johansson’s stronger result [86] that the same bound holds even for triangle-free graphs.
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© 2002 Springer-Verlag Berlin Heidelberg
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Molloy, M., Reed, B. (2002). Triangle-Free Graphs. In: Graph Colouring and the Probabilistic Method. Algorithms and Combinatorics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04016-0_13
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DOI: https://doi.org/10.1007/978-3-642-04016-0_13
Publisher Name: Springer, Berlin, Heidelberg
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