ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 21(1) (2001) 267-281
DOI: 10.7151/dmgt.1149


Bert Randerath

Institut für Informatik
Universität zu Köln
D-50969 Köln, Germany

Ingo Schiermeyer

Fakultät für Mathematik und Informatik
TU Bergakademie Freiberg
D-09596 Freiberg, Germany


In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P5-free or triangle-free, P6-free and C6-free graphs are 3-colourable. A canonical extension of these graph classes is GI(4, 5), the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of GI(4,5) are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have polynomial time algorithms to 3-colour every G ∈ GI(n1,n2) with n1,n2 ≥ 4 (see Table 1). Furthermore, we consider the related problem of finding a χ-binding function for the class GI(n1,n2). Here we obtain the surprising result that there exists no linear χ-binding function for GI(3,4).

Keywords: colouring, chromatic number, induced subgraphs, graph algorithms, χ-binding function.

2000 Mathematics Subject Classification: 05C15, 05C75, 05C85.


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Received 28 December 2000
Revised 13 May 2001