A 4-colour problem for dense triangle-free graphs

Authors:
Stephan Brandt
Affiliations:
Technische Universität Ilmenau, Fak. für Mathematik & Naturwissenschaften, Postfach 100 565, D-98684 Ilmenau, Germany
Venue:
Discrete Mathematics
Year:
2002

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Abstract

Triangle-free graphs of order n with minimum degree exceeding n/3 satisfy strong structural properties in several respects. Nevertheless, it is not known whether those graphs can have arbitrarily large chromatic number. We conjecture that these graphs are 4-colourable and give an affirmative answer for regular maximal triangle-free graphs satisfying the degree bound. Moreover, we show that the vertex transitive members of this class are 3-colourable. The indicated problem has a fractional appeal and we present several related conjectures in fractional terms.